Celebratio Mathematica

Ad­ri­an Al­bert, one of the fore­most al­geb­ra­ists of the world and Pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety from 1965 to 1967, died on June 6, 1972. For al­most a year be­fore his death it had be­come ap­par­ent to his friends that his man­ner had altered from its cus­tom­ary vig­or to one which was rather sub­dued. At first they at­trib­uted this to a let­down which might have res­ul­ted from Al­bert’s hav­ing re­cently re­lin­quished a very de­mand­ing ad­min­is­trat­ive po­s­i­tion (Dean of the Di­vi­sion of Phys­ic­al Sci­ences at the Uni­versity of Chica­go) that he had held for a num­ber of years. Even­tu­ally it be­came known that he was gravely ill of phys­ic­al causes that had their ori­gin in dia­betes with which he had been af­flic­ted for many years.

Al­bert was a first gen­er­a­tion Amer­ic­an and a second gen­er­a­tion Amer­ic­an math­em­atician fol­low­ing that of E. H. Moore, Os­wald Veblen, L. E. Dick­son and G. D. Birk­hoff. His moth­er came to the United States from Kiev and his fath­er came from Eng­land.1 The fath­er had run away from his home in Vilna at the age of four­teen, and on ar­riv­ing in Eng­land, he dis­carded his fam­ily name (which re­mains un­known) and took in its place the name Al­bert after the prince con­sort of Queen Vic­tor­ia. Al­bert’s fath­er was something of a schol­ar, with a deep in­terest in Eng­lish lit­er­at­ure. He taught school for a while in Eng­land but after com­ing to the United States he be­came a sales­man, a shop­keep­er, and a man­u­fac­turer. Ad­ri­an was born when his fath­er was fifty-five and his moth­er was thirty-five. It was a second mar­riage for both par­ents; his fath­er’s first wife had died in child­birth, and his moth­er was a wid­ow with two chil­dren when she mar­ried his fath­er. Ad­ri­an was the middle child of a set of three chil­dren which his par­ents had in com­mon. He grew up in a fam­ily that was form­ally or­tho­dox Jew­ish but not strongly re­li­gious. In com­mon with most im­mig­rant fam­il­ies of the peri­od the fam­ily had a strong drive to­ward as­sim­il­a­tion and a de­term­in­a­tion to make the most of the op­por­tun­it­ies offered by a com­par­at­ively free so­ci­ety un­der­go­ing rap­id eco­nom­ic ex­pan­sion with no lim­its in sight.

Al­bert spent all of his school years in the Mid­w­est and all but two of these in Chica­go. He at­ten­ded pub­lic schools at Chica­go and at Iron Moun­tain, Michigan, and entered the Uni­versity of Chica­go in 1922 where in rap­id suc­ces­sion he earned a B.S. de­gree in 1926, an M.S. de­gree in 1927, and a Ph.D. in 1928. His ad­visor for his mas­ter’s and his doc­tor­al dis­ser­ta­tions was Le­onard Eu­gene Dick­son. After his doc­tor­ate Al­bert spent a year at Prin­ceton Uni­versity as a Na­tion­al Re­search Coun­cil Fel­low. He was at­trac­ted to Prin­ceton by that great mas­ter of as­so­ci­at­ive al­gebra the­ory, J. H. M. Wed­der­burn, who was then a pro­fess­or at the uni­versity. Al­bert re­turned to Prin­ceton in 1933, this time as one of the first group of tem­por­ary mem­bers of the In­sti­tute for Ad­vanced Study.

Al­bert mar­ried Frieda Dav­is in 1927, and they had three chil­dren, Alan, Roy, and Nancy, one of whom, Roy, died of dia­betes at the age of twenty-three.

Ex­cept for two years (1929–1931) as an In­struct­or at Columbia Uni­versity and a num­ber of vis­it­ing pro­fess­or­ships (at Rio de Janeiro, Buenos Aires, Uni­versity of South­ern Cali­for­nia, Yale, and the Uni­versity of Cali­for­nia at Los Angeles) all of Al­bert’s aca­dem­ic ca­reer was spent at the Uni­versity of Chica­go. In 1960 he was named Eliakim Hast­ings Moore Dis­tin­guished Ser­vice Pro­fess­or, and he served as Chair­man of the De­part­ment of Math­em­at­ics for three years un­til he be­came Dean of the Di­vi­sion of Phys­ic­al Sci­ences in 1962. He held this po­s­i­tion un­til 1971 when he reached the man­dat­ory re­tire­ment age of sixty-five for the dean­ship.

Of the math­em­aticians who in­flu­enced Al­bert most dir­ectly we should list the fol­low­ing: Dick­son, who set the dir­ec­tion for al­most all of Al­bert’s re­search and whose books, Al­geb­ras and their Arith­met­ics (1923) and Al­gebren und ihre Zah­len­the­or­ie (1927), stim­u­lated the great flower­ing of as­so­ci­at­ive al­gebra the­ory of the 1930’s; Wed­der­burn, whose el­eg­ant res­ults and meth­ods were an in­spir­a­tion to Al­bert; Her­mann Weyl, whose lec­tures on Lie groups and es­pe­cially Lie al­geb­ras aroused Al­bert’s in­terest in this sub­ject — an in­terest which later broadened to en­com­pass the whole range of nonas­so­ci­at­ive al­geb­ras; and above all, So­lomon Lef­schetz, who in­tro­duced Al­bert to the sub­ject of Riemann matrices dur­ing his postdoc­tor­al year (1928–1929) at Prin­ceton.

Mrs. Al­bert tells the story of this in­tro­duc­tion in a charm­ing fash­ion. Filling in some math­em­at­ic­al de­tails it runs some­what as fol­lows. Al­bert had giv­en a lec­ture on his dis­ser­ta­tion at the Prin­ceton math­em­at­ics club. In the audi­ence were Dieud­on­né, J. H. C. White­head and Lef­schetz, who had worked on the prob­lem of mul­ti­plic­a­tion al­geb­ras of Riemann matrices. Lef­schetz ap­par­ently sensed that here was a bril­liant young al­geb­ra­ist whose in­terests and power made him ideally suited to at­tack this prob­lem. After Al­bert’s talk he de­scribed the prob­lem to him. A lively dis­cus­sion en­sued, mostly in the course of wan­der­ings through the streets of Prin­ceton. This las­ted for sev­er­al hours, well past din­ner­time, and Mrs. Al­bert had be­come quite con­cerned be­fore Al­bert fi­nally re­turned home, ap­par­ently in great ex­cite­ment over his ini­ti­ation in­to a fas­cin­at­ing area of clas­sic­al math­em­at­ics which provided a strong mo­tiv­a­tion for the study of his chosen field of as­so­ci­at­ive al­geb­ras.

Lef­schetz was cer­tainly right in his judg­ment. Al­bert took to the prob­lem on Riemann matrices with great en­thu­si­asm, and as the struc­ture the­ory of as­so­ci­at­ive al­geb­ras was re­vealed by Al­bert, Brauer, Hasse and Emmy No­eth­er, Al­bert could push for­ward the the­ory of mul­ti­plic­a­tion al­geb­ras of Riemann matrices un­til he achieved a com­plete solu­tion of the cent­ral prob­lem (which we shall dis­cuss be­low). For this achieve­ment Al­bert was awar­ded the Cole Prize in al­gebra in 1939.

This was a mem­or­able year for Al­bert. Be­sides the Cole Prize award which he re­ceived that year, he was the Col­loqui­um speak­er of the So­ci­ety for 1939. Moreover, he per­formed a feat, which we be­lieve has nev­er been matched, of hav­ing the book, Struc­ture of Al­geb­ras, the sub­ject of his lec­tures in print at the same time that the lec­tures were de­livered.

Around 1942 Al­bert’s re­search in­terests shif­ted from as­so­ci­at­ive to nonas­so­ci­at­ive al­geb­ras. He wrote many im­port­ant pa­pers in this field (which we shall dis­cuss be­low). In 1965 Al­bert re­turned to his first love, struc­ture the­ory of as­so­ci­at­ive al­geb­ras.

Be­sides his own im­port­ant con­tri­bu­tions to math­em­at­ics, Al­bert was in­stru­ment­al in a num­ber of ways in im­prov­ing the status of the pro­fes­sion. He had a good deal to do with the es­tab­lish­ment of gov­ern­ment re­search grants for math­em­at­ics on more or less an equal foot­ing with those in the oth­er sci­ences. He was chair­man of the Com­mit­tee to Pre­pare a Budget for Math­em­at­ics for the Na­tion­al Sci­ence Found­a­tion, 1950, and chair­man of the Com­mit­tee on a Sur­vey of Train­ing and Re­search Po­ten­tial in the Math­em­at­ic­al Sci­ences, Janu­ary 1955–June 1957 (which be­came known as “The Al­bert Com­mit­tee”). He demon­strated that pure math­em­aticians could be use­ful in ap­plied and dir­ec­ted re­search by act­ing as a con­sult­ant for a num­ber of gov­ern­ment sponsored re­search agen­cies. For a num­ber of years he was as­so­ci­ated with the In­sti­tute for De­fense Ana­lys­is as a mem­ber of its Board of Trust­ees and for a year as Dir­ect­or of its Prin­ceton group. He dir­ec­ted the re­search pro­ject SCAMP for sev­er­al sum­mers and or­gan­ized and dir­ec­ted the pro­ject ALP (known as “Al­bert’s little pro­ject”).

Al­bert was also a driv­ing force in the cre­ation of the sum­mer re­search in­sti­tutes which have be­come such an im­port­ant part of the re­search activ­it­ies of the So­ci­ety, sup­por­ted by the Na­tion­al Sci­ence Found­a­tion. He was chair­man of the com­mit­tee which was re­spons­ible for the first one of these — on Lie groups and Lie al­geb­ras — held at Colby Col­lege in Maine in the sum­mer of 1953.

Al­bert’s role as a “states­man” for math­em­at­ics in­cluded mem­ber­ship on the Board of Trust­ees of the In­sti­tute for Ad­vanced Study, chair­man of the Con­sultat­ive Com­mit­tee of the Nice Con­gress, and Vice-Pres­id­ent of the In­ter­na­tion­al Math­em­at­ic­al Uni­on.

His in­flu­ence in math­em­at­ics ex­ten­ded also through a large num­ber of gif­ted stu­dents. One of the most dis­tin­guished of these, Dan Zel­in­sky, has writ­ten a warm ap­pre­ci­ation of Al­bert as a math­em­atician and as a per­son [e37].

Nat­ur­ally many im­port­ant hon­ors came his way. He was elec­ted to the Na­tion­al Academy of Sci­ences in 1943 and was awar­ded hon­or­ary doc­tor­ates from Notre Dame, Ye­shiva Uni­versity, and the Uni­versity of Illinois. He was elec­ted a cor­res­pond­ing mem­ber of the Brazili­an Academy of Sci­ences, hon­or­ary mem­ber of the Ar­gen­tine Academy of Sci­ences, and of the Mex­ic­an Math­em­at­ic­al So­ci­ety. He thor­oughly en­joyed these hon­ors, but he de­rived al­most as much pleas­ure from the hon­ors be­stowed on fel­low al­geb­ra­ists and on his friends. Most of all he en­joyed seek­ing out a col­league to whom he could com­mu­nic­ate his latest dis­cov­ery, which ex­cited him greatly.

Most of Al­bert’s im­port­ant dis­cov­er­ies fall neatly in­to three cat­egor­ies: I. As­so­ci­at­ive al­geb­ras, II. Riemann matrices, III. Nonas­so­ci­at­ive al­geb­ras. We pro­ceed to give an in­dic­a­tion of these and of some in­ter­est­ing isol­ated res­ults which we shall men­tion un­der IV. Mis­cel­laneous.

The Wed­der­burn struc­ture the­or­ems of 1907 on fi­nite di­men­sion­al as­so­ci­at­ive al­geb­ras over a field fo­cused at­ten­tion on the di­vi­sion al­geb­ras in this class. In 1906 Dick­son had giv­en a con­struc­tion of a type of al­gebra called cyc­lic which in­cluded di­vi­sion al­geb­ras. These con­tain a max­im­al sub­field $\mathfrak{Z}$ which is cyc­lic over the base field $\mathfrak{F}$, that is, they are Galois with Galois group $G=⟨s⟩$, a cyc­lic group gen­er­ated by a single ele­ment $s$. Moreover, the al­geb­ras are gen­er­ated by $\mathfrak{Z}$ and an ele­ment $u$ for which one has the re­la­tions $$uz=s(z)u,z\in \mathfrak{Z},u^n=\gamma ,$$ where $n$ is the or­der of $G$ and $\gamma$ is a nonzero ele­ment of $\mathfrak{F}$. The cyc­lic al­gebra, de­noted as $(\mathfrak{Z},s,\gamma )$, con­struc­ted in this way has di­men­sion­al­ity $n^2$ over $\mathfrak{F}$. In [e1] Wed­der­burn proved an im­port­ant suf­fi­cient con­di­tion for $(\mathfrak{Z},s,\gamma )$ to be a di­vi­sion al­gebra. He showed that this is the case if no power of $\gamma ,{\gamma }^m$ with $0<m<n$, is a norm $N_{\mathfrak{Z}/\mathfrak{F}}(z)$ of an ele­ment $z\in \mathfrak{Z}$. Us­ing this cri­terion it is easy to con­struct di­vi­sion al­geb­ras of any di­men­sion $n^2$.

In 1921 Wed­der­burn pub­lished some oth­er im­port­ant res­ults on di­vi­sion al­geb­ras [e2]. Not­ing that one may as well con­sider these as al­geb­ras over their cen­ters and so as­sume that they are cent­ral in the sense that the cen­ter is the base field $\mathfrak{F}$, he showed that the di­men­sion­al­ity over this field is a square, $n^2$. More gen­er­ally, if $\mathfrak{U}$ is cent­ral simple, by one of Wed­der­burn’s struc­ture the­or­ems, $\mathfrak{U}$ is the al­gebra $M_r(\mathfrak{D})$ of $r\times r$ matrices with ele­ments in a cent­ral di­vi­sion al­gebra $\mathfrak{D}$. Hence if the di­men­sion­al­ity of $\mathfrak{D}$ over $\mathfrak{F}$ is $d^2$, then that of $\mathfrak{U}$ over $\mathfrak{F}$ is $n^2$, where $n=dr$. Then $n$ is called the de­gree of the cent­ral simple al­gebra $\mathfrak{U}$ and $d$ is its in­dex. In his 1921 pa­per, Wed­der­burn showed also that any max­im­al sub­field $\mathfrak{Z}$ of a cent­ral di­vi­sion al­gebra $\mathfrak{D}$ is a split­ting field, that is, the al­gebra ${\mathfrak{D}}^{\mathfrak{Z}}=\mathfrak{Z}{\otimes }_{\mathfrak{F}}\mathfrak{D}$ is the mat­rix al­gebra $M_d(\mathfrak{Z})$, and he proved that every cent­ral di­vi­sion al­gebra of de­gree three is cyc­lic. Wed­der­burn showed also that Dick­son’s cyc­lic al­geb­ras were spe­cial cases of a more gen­er­al type of al­gebra which is now called an abeli­an crossed product. Here the cyc­lic field $\mathfrak{Z}$ is re­placed by a Galois ex­ten­sion field of the base field with Galois group an abeli­an group.

Abeli­an crossed products were re­dis­covered by Cecioni [e3], and these were fur­ther gen­er­al­ized by Dick­son [e4] and [e6] to ar­bit­rary crossed products based on any Galois ex­ten­sion field.

Much of Al­bert’s early work was con­cerned with the study of fi­nite di­men­sion­al cent­ral simple al­geb­ras. His first im­port­ant res­ult on these was the the­or­em, proved in his dis­ser­ta­tion [1], that every cent­ral di­vi­sion al­gebra of de­gree four (di­men­sion six­teen) is a crossed product. This was the next case to be con­sidered after Wed­der­burn’s the­or­em that in de­gree three these al­geb­ras are cyc­lic. Al­bert im­proved the res­ult in [2] by show­ing that the de­gree four cent­ral di­vi­sion al­geb­ras are crossed products based on abeli­an ex­ten­sion fields whose Galois groups are dir­ect products of two cyc­lic groups of or­der two, and he gave a sim­pler proof of this res­ult in [5]. In both of these pa­pers the al­geb­ras of char­ac­ter­ist­ic two were ex­cluded. In a sub­sequent pa­per [12] he was able to over­come the dif­fi­culties of the char­ac­ter­ist­ic two case. Brauer was the first to show that the cent­ral di­vi­sion al­geb­ras of de­gree four, un­like those of de­gree three, need not be cyc­lic. He con­struc­ted an ex­ample of such an al­gebra which was a tensor product of two (gen­er­al­ized) qua­ternion al­geb­ras [e7]. Sub­sequently, Al­bert [6] con­struc­ted one which is not such a product. This was sig­ni­fic­ant in view of an­oth­er im­port­ant the­or­em, proved by Al­bert [34], stat­ing that a cent­ral di­vi­sion al­gebra $\mathfrak{D}$ of de­gree four is a tensor product of qua­ternion al­geb­ras if and only if $\mathfrak{D}{\otimes }_{\mathfrak{F}}\mathfrak{D}\cong M_4(\mathfrak{F})$.

The main goal of the struc­ture the­ory of al­geb­ras of the peri­od 1929–1932 was the de­term­in­a­tion and clas­si­fic­a­tion of fi­nite di­men­sion­al di­vi­sion al­geb­ras over the field $\mathit{Q}$ of ra­tion­al num­bers, or equi­val­ently, fi­nite di­men­sion­al cent­ral di­vi­sion al­geb­ras over num­ber fields. It was re­cog­nized quite early that this prob­lem had two sep­ar­ate as­pects: a purely al­geb­ra­ic one con­cerned with prop­er­ties of al­geb­ras val­id for all base fields, and an arith­met­ic one ex­ploit­ing the arith­met­ic of num­ber fields. Al­bert re­cog­nized the im­port­ance of the arith­met­ic meth­od. However, he was han­di­capped in its use by the fact that he was un­aware un­til rather late of the power­ful res­ults of al­geb­ra­ic num­ber the­ory, not­ably, class field the­ory, which had been de­veloped in Ger­many. He did make use of the arith­met­ic the­ory of quad­rat­ic forms to achieve defin­it­ive res­ults on cent­ral di­vi­sion al­geb­ras of de­gree four over num­ber fields and some im­port­ant early res­ults on the de­gree $2^n$ case. For ex­ample, he proved that the former are cyc­lic and are not tensor products of qua­ternion al­geb­ras, and he proved that the only cent­ral di­vi­sion al­geb­ras over num­ber fields which pos­sess in­vol­u­tions, that is, an­ti­auto­morph­isms of peri­od two, are the qua­ternion al­geb­ras. This last res­ult was needed for his study of Riemann matrices which we shall dis­cuss be­low.

Al­bert’s main con­tri­bu­tions were on the purely al­geb­ra­ic side. There is a sub­stan­tial over­lap between his res­ults on cent­ral simple al­geb­ras and those of the Ger­man school of al­geb­ra­ists of the peri­od of the early thirties, es­pe­cially those of Richard Brauer and of Emmy No­eth­er. Al­bert ob­tained in­de­pend­ently all the al­geb­ra­ic res­ults on split­ting fields, ex­ten­sions of iso­morph­isms and tensor products which were needed to ob­tain the fun­da­ment­al the­or­ems on di­vi­sion al­geb­ras over num­ber fields. Of cent­ral im­port­ance for the al­geb­ra­ic the­ory is the group of classes of cent­ral simple al­geb­ras which was in­tro­duced by Brauer in 1929 [e7]. We re­call the defin­i­tion. Two (fi­nite di­men­sion­al) cent­ral simple al­geb­ras $\mathfrak{U}$ and $\mathfrak{B}$ over a field $\mathfrak{F}$ are said to be sim­il­ar ($\sim$) if there ex­ist pos­it­ive in­tegers $m$ and $n$ such that the mat­rix al­geb­ras $M_m(\mathfrak{U})$ and $M_n(\mathfrak{B})$ are iso­morph­ic. This is an equi­val­ence re­la­tion. De­not­ing the sim­il­ar­ity class of $\mathfrak{U}$ as $\{\mathfrak{U}\}$, one defines a product of such classes by $\{\mathfrak{U}\}\{\mathfrak{B}\}=\{\mathfrak{U}{\otimes }_{\mathfrak{F}}\mathfrak{B}\}$. This gives a com­mut­at­ive group $B(\mathfrak{F})$ called the Brauer group of the field $\mathfrak{F}$. The unit of the group is the set of mat­rix al­geb­ras $M_n(\mathfrak{F})$, $n=1,2,\dots$, and the in­verse of $\{\mathfrak{U}\}$ is $\{{\mathfrak{U}}^{\mathrm{op}}\}$, where ${\mathfrak{U}}^{\mathrm{op}}$ is the op­pos­ite al­gebra of $\mathfrak{U}$. In a beau­ti­ful pa­per [3] pub­lished in 1931, Al­bert es­sen­tially re­dis­covered the Brauer group. In this pa­per he proved Brauer’s main the­or­em that $B(\mathfrak{F})$ is a tor­sion group; more pre­cisely, if $\mathfrak{U}$ has in­dex $m$, that is, if the de­gree of the di­vi­sion al­gebra $\mathfrak{D}$ in $\{\mathfrak{U}\}$ is $m$, then $\{\mathfrak{U}{\}}^m=1$. Moreover, if $e$ is the or­der of $\{\mathfrak{U}\}$ in $B(\mathfrak{F})$, then $e$ and $m$ have the same prime factors. The in­teger $e$ is called the ex­po­nent of $\mathfrak{U}$. Al­bert’s proofs are based on Wed­der­burn’s norm con­di­tion for cyc­lic al­geb­ras to be di­vi­sion al­geb­ras and the­or­ems re­du­cing con­sid­er­a­tions to the cyc­lic case; for ex­ample, if $\mathfrak{D}$ is a cent­ral di­vi­sion al­gebra of prime de­gree $p$, then there ex­ists an ex­ten­sion field $\mathfrak{R}$ of the base field of di­men­sion­al­ity prime to $p$ such that ${\mathfrak{D}}^{\mathfrak{R}}=\mathfrak{R}{\otimes }_{\mathfrak{F}}\mathfrak{D}$ is a cyc­lic di­vi­sion al­gebra over $\mathfrak{R}$. An­oth­er key tool in Al­bert’s meth­od was the fol­low­ing the­or­em which he called the in­dex re­duc­tion factor the­or­em: Let $\mathfrak{D}$ be a cent­ral di­vi­sion al­gebra of de­gree $d$ and $\mathfrak{R}$ an ex­ten­sion field of the base field $\mathfrak{F}$ with di­men­sion­al­ity $r$. Then ${\mathfrak{D}}^{\mathfrak{R}}=M_q(\mathfrak{E})$, where $\mathfrak{E}$ is a cent­ral di­vi­sion al­gebra over $\mathfrak{R}$ and $q$ is a di­visor of $d$ and $r$. Al­bert’s primary in­terest in the the­or­em that $\{\mathfrak{U}{\}}^m=1$ was its con­sequence that any cent­ral di­vi­sion al­gebra is a tensor product of di­vi­sion al­geb­ras of prime power de­grees which are de­term­ined up to iso­morph­ism. This re­duced most ques­tions on these al­geb­ras to the prime power de­gree case.

The high points of the struc­ture the­ory of al­geb­ras of the 1930’s were un­doubtedly the the­or­em that every fi­nite di­men­sion­al cent­ral di­vi­sion al­gebra over a num­ber field is cyc­lic, and the clas­si­fic­a­tion of these al­geb­ras by a set of nu­mer­ic­al in­vari­ants. The lat­ter res­ult amounts to the de­term­in­a­tion of the struc­ture of the Brauer group for a num­ber field. Be­sides the gen­er­al the­ory of cent­ral simple al­geb­ras we have in­dic­ated, the proofs of these fun­da­ment­al res­ults re­quired the struc­ture the­ory of cent­ral simple al­geb­ras over $p$-ad­ic fields due to Hasse, Hasse’s norm the­or­em (“the Hasse prin­ciple”), and the Grun­wald ex­ist­ence the­or­em for cer­tain cyc­lic ex­ten­sions of a num­ber field. (Though it was dis­covered al­most thirty years later by S. Wang [e19] that Grün­wald’s for­mu­la­tion was in­cor­rect, his er­ror did not af­fect the the­or­em on al­geb­ras. See also Wang [e21] and Hasse [e22].) The first proof of the cyc­lic struc­ture of cent­ral di­vi­sion al­geb­ras over num­ber fields was giv­en by Brauer, Hasse and No­eth­er [e8]. However, it seemed ap­pro­pri­ate that Al­bert should share the hon­or of this achieve­ment, and at Hasse’s sug­ges­tion a joint pa­per ([4], 1932) was pub­lished by Al­bert and Hasse giv­ing an­oth­er proof of the the­or­em and the his­tor­ic­al back­ground of the prob­lem.

The res­ults which had been ob­tained up to this point sug­ges­ted the fol­low­ing two prob­lems: (I) Is every fi­nite di­men­sion­al cent­ral di­vi­sion al­gebra a crossed product? (II) Is every one of prime de­gree cyc­lic? These are equi­val­ent to the ques­tion of ex­ist­ence of a max­im­al Galois and max­im­al cyc­lic sub­field, re­spect­ively, for these al­geb­ras. The res­ults of Wed­der­burn and Al­bert im­ply that the an­swer to the second ques­tion is af­firm­at­ive for the primes 2 and 3 and for the first for the de­grees 2, 3, 4, 6 and 12. Quite re­cently Amit­sur showed that the an­swer to the first ques­tion is neg­at­ive by show­ing that for any $n$ di­vis­ible by eight or by the square of an odd prime there ex­ists a non­crossed product cent­ral di­vi­sion al­gebra of de­gree $n$ [e36]. This leaves in­tact the second prob­lem, and this is one on which Al­bert spent a good deal of ef­fort. It is clear from the defin­i­tion that if $\mathfrak{U}$ is cyc­lic of de­gree $n$, then $\mathfrak{U}$ con­tains an ele­ment $u$ sat­is­fy­ing an ir­re­du­cible pure equa­tion $x^n-\gamma =0$, $\gamma$ in the base field $\mathfrak{F}$. Does the con­verse hold? Al­bert showed this is the case if $n=p$, a prime [8], and is not the case if $n=4$ [22], For the prime case this re­duces the prob­lem (II) to what ap­pears to be a more tract­able one: Does every cent­ral di­vi­sion al­gebra $\mathfrak{U}$ of prime de­gree $p$ con­tain an ele­ment not in $\mathfrak{F}$ whose $p$-th power is in $\mathfrak{F}$? In 1938 Brauer showed that if $\mathfrak{U}$ is of de­gree 5 there ex­ists a field $\mathfrak{R}$ con­tain­ing a tower of fields $\mathfrak{F}\subset {\mathfrak{R}}_1\subset {\mathfrak{R}}_2\subset \mathfrak{R}$ such that the de­gree $[{\mathfrak{R}}_1:\mathfrak{F}]=2=[{\mathfrak{R}}_2:{\mathfrak{R}}_1]$ and $[\mathfrak{R}:{\mathfrak{R}}_2]=3$, and ${\mathfrak{U}}^{\mathfrak{R}}$ is cyc­lic [e16]. This led Al­bert to con­sider the fol­low­ing ques­tion: Sup­pose $\mathfrak{R}$ is a quad­rat­ic ex­ten­sion of $\mathfrak{F}$ and ${\mathfrak{U}}^{\mathfrak{R}}$ is cyc­lic of prime de­gree. Then is $\mathfrak{U}$ cyc­lic? In four pa­pers ap­pear­ing between 1965 and 1970, in­clud­ing his re­tir­ing Pres­id­en­tial Ad­dress for the So­ci­ety, Al­bert con­sidered this prob­lem for al­geb­ras of char­ac­ter­ist­ic $p$ and de­gree $p$ [48], [50], [51], [52]. In spite of many in­geni­ous ar­gu­ments and par­tial res­ults, he was un­able to com­pletely settle this ques­tion.

A beau­ti­ful chapter in the struc­ture the­ory of cent­ral simple al­geb­ras is the the­ory of $p$-al­geb­ras which Al­bert de­veloped in three pa­pers ap­pear­ing in 1936 and 1937 [18], [19], [20] (cf. also [e12]). These are the cent­ral simple al­geb­ras of char­ac­ter­ist­ic $p$ whose di­vi­sion al­geb­ras $\mathfrak{D}$ in the Wed­der­burn the­or­em ($\cong M_n(\mathfrak{D})$) have de­gree a power of $p$. The main res­ults Al­bert proved about $p$-al­geb­ras are that any such al­gebra $\mathfrak{U}$ is cyc­lic­ally rep­res­ent­able, that is, there ex­ists an $n$ such that $M_n(\mathfrak{U})$ is a cyc­lic al­gebra, and the ex­po­nent of $\mathfrak{U}$ is the min­im­um of the ex­po­nents of purely in­sep­ar­able split­ting fields for the al­gebra.

A gen­er­al­iz­a­tion of cyc­lic al­geb­ras in which the cyc­lic max­im­al sub­field $\mathfrak{Z}$ is re­placed by a sep­ar­able com­mut­at­ive sub­al­gebra on which a cyc­lic group $G$ acts in such a way that there are no prop­er sub­al­geb­ras sta­bil­ized by $G$ was con­sidered by Al­bert in [21] fol­low­ing earli­er work by Teich­müller in [e13]. Such gen­er­al­ized cyc­lic al­geb­ras arise nat­ur­ally from cyc­lic ones when one ex­tends the base field or forms the tensor powers of a cyc­lic al­gebra.

Most of the im­port­ant res­ults on as­so­ci­at­ive al­geb­ras which Al­bert ob­tained pri­or to 1939 can be found in an im­proved form in his AMS Col­loqui­um book, Struc­ture of Al­geb­ras. This ex­tremely read­able and beau­ti­fully or­gan­ized book can still be re­com­men­ded to a be­gin­ning stu­dent with a ser­i­ous in­terest in struc­ture the­ory and is an in­dis­pens­able ref­er­ence book for cer­tain as­pects of the the­ory, par­tic­u­larly the the­ory of $p$-al­geb­ras, and of al­geb­ras with in­vol­u­tion.

The the­ory of mul­ti­plic­a­tions of Riemann matrices has its ori­gin in al­geb­ra­ic geo­metry. On a Riemann sur­face of an al­geb­ra­ic curve of genus $p$, one chooses $p$ lin­early in­de­pend­ent in­teg­rals of the first kind each with $2p$ peri­ods ${\omega }_{j\nu }$, $1\leqq j\leqq p$, $1\leqq \nu \leqq 2p$. The $p\times 2p$ mat­rix $\omega =({\omega }_{j\nu })$ of com­plex ele­ments sat­is­fies the Riemann re­la­tions: there ex­ists a $2p\times 2p$ nonsin­gu­lar skew sym­met­ric mat­rix $C$ of ra­tion­al ele­ments such that $\omega C{}^t\omega =0$ (${}^t\omega$ the trans­pose of $\omega$) and $\sqrt{-1}\omega C{}^t\overline{\omega }$ is pos­it­ive def­in­ite her­mitian. In the the­ory of so-called sin­gu­lar cor­res­pond­ences on the Riemann sur­face, one is led to con­sider the mul­ti­plic­a­tions of $\omega$. These are the $2p\times 2p$ ra­tion­al matrices $A$ for which there ex­ists $p\times p$ com­plex mat­rix $\alpha$ such that $\alpha \omega =\omega A$. The set of these $A$’s is a fi­nite di­men­sion­al al­gebra of matrices over $\mathit{Q}$, the al­gebra of mul­ti­plic­a­tions of $\omega$.

Al­tern­at­ively, the matrices $\omega$ and their mul­ti­plic­a­tions arise in the the­ory of abeli­an func­tions, defined to be mero­morph­ic func­tions of $p$ com­plex vari­ables hav­ing a lat­tice of peri­ods in ${\mathit{C}}^p$.

There is an­oth­er, form­ally sim­pler, for­mu­la­tion of Riemann matrices (the fore­go­ing $\omega$) and their mul­ti­plic­a­tions due to Weyl [e10] which was sug­ges­ted by geo­met­ric con­sid­er­a­tions. From the purely form­al point of view one ob­tains the pas­sage from the clas­sic­al for­mu­la­tion to Weyl’s by in­tro­du­cing the $2p\times 2p$ mat­rix $$W=(\genfrac{}{}{0ex}{}{\omega }{\overline{\omega }})\text{and}L=\left(\begin{array}{cc}-\sqrt{-1}{\mathbf{1}}_p& 0\\ 0& \sqrt{-1}{\mathbf{1}}_p\end{array}\right).$$ Put $R=W^{-1}LW$. Then it can be shown that the mat­rix $R$ has the fol­low­ing prop­er­ties: (1) $R$ is real, that is $R\in M_{2p}(\mathit{R})$; (2) $R^2=-{\mathbf{1}}_{2p}$; (3) $S=CR$ is pos­it­ive def­in­ite sym­met­ric. Fol­low­ing Weyl, one calls a mat­rix $R\in M_n(\mathit{R})$ (here $n=2p$) a Riemann mat­rix if $R^2=-{\mathbf{1}}_n$ and there ex­ists a skew sym­met­ric mat­rix $C\in M_n(\mathit{Q})$ such that $S=CR$ is pos­it­ive def­in­ite sym­met­ric. The mat­rix $C$, which is ne­ces­sar­ily nonsin­gu­lar, is called a prin­cip­al mat­rix of $R$. The pas­sage from Weyl’s $R$ to the clas­sic­al $\omega$ can be re­versed. In Weyl’s for­mu­la­tion the mul­ti­plic­a­tions ap­pear as the matrices $A\in M_n(\mathit{Q})$ com­mut­ing with $R$. The set $\mathfrak{U}$ of these mul­ti­plic­a­tions is a fi­nite di­men­sion­al al­gebra over $\mathit{Q}$ called the mul­ti­plic­a­tion al­gebra of the Riemann mat­rix $R$. Weyl ob­served that for most con­sid­er­a­tions the con­di­tion $R^2=-\mathbf{1}$ $(=-{\mathbf{1}}_n)$ plays no role. Drop­ping this, one ob­tains gen­er­al­ized Riemann matrices. Sub­sequently Al­bert [23], [17] con­sidered fur­ther gen­er­al­iz­a­tions (in­clud­ing even a char­ac­ter­ist­ic $p\ne 0$ situ­ation!). For the sake of sim­pli­city we shall stick to the case of Riemann matrices in Weyl’s for­mu­la­tion.

The im­port­ant early work on mul­ti­plic­a­tion al­geb­ras is due to Poin­caré, Scorza, Lef­schetz and Ros­ati. Poin­caré achieved a re­duc­tion to so-called pure Riemann matrices for which the mul­ti­plic­a­tion al­geb­ras are di­vi­sion al­geb­ras. Lef­schetz con­sidered the situ­ation in which the mul­ti­plic­a­tion al­geb­ras are com­mut­at­ive. Ros­ati ob­served the im­port­ant fact that if $A$ is in the mul­ti­plic­a­tion al­gebra $\mathfrak{U}$ of a Riemann mat­rix $R$ and $C$ is a prin­cip­al mat­rix, then $A^{\ast }=C^{-1}{}^{t}AC\in \mathfrak{U}$. The map $A\to A^{\ast }$ is an in­vol­u­tion (an­ti­auto­morph­ism of peri­od two) in $\mathfrak{U}$. Ros­ati showed also that if $A$ is sym­met­ric un­der this in­vol­u­tion $(A^{\ast }=A)$ then its char­ac­ter­ist­ic roots are real, and if $A^{\ast }=-A$ then its char­ac­ter­ist­ic roots are pure ima­gin­ar­ies [e5].

The cent­ral prob­lem on mul­ti­plic­a­tion al­geb­ras of Riemann matrices is to de­term­ine ne­ces­sary and suf­fi­cient con­di­tions that a di­vi­sion al­gebra over $\mathit{Q}$ be the mul­ti­plic­a­tion al­gebra of a Riemann mat­rix. For a proof of suf­fi­ciency, one re­quires a con­struc­tion of a Riemann mat­rix whose mul­ti­plic­a­tion al­gebra is a giv­en al­gebra $\mathfrak{U}$ sat­is­fy­ing the con­di­tions.

Al­bert’s work on Riemann matrices went hand in hand with the de­vel­op­ment of the the­ory of di­vi­sion al­geb­ras. It cul­min­ated in the com­plete solu­tion of the prin­cip­al prob­lem, which he pub­lished in three pa­pers ap­pear­ing in the An­nals of Math­em­at­ics in 1934 and 1935 ( [9], [11] and [16]). To achieve this re­quired the de­vel­op­ment ab ini­tio of the ba­sic the­ory of simple al­geb­ras with in­vol­u­tion. Al­bert presen­ted im­proved ver­sions of this the­ory in [17] and in his Struc­ture of Al­geb­ras. We shall in­dic­ate first his res­ults on al­geb­ras with in­vol­u­tion.

We as­sume throughout that $\mathfrak{U}$ is fi­nite di­men­sion­al simple over a field $\mathfrak{F}$. If $\mathfrak{U}$ has an in­vol­u­tion $J$ ($J:a\to a^{\ast }$ such that $(a+b{)}^{\ast }=a^{\ast }+b^{\ast }$, $(\alpha a{)}^{\ast }=\alpha a^{\ast }$ for $\alpha \in \mathfrak{F}$, $(ab{)}^{\ast }=b^{\ast }{a}^{\ast }$), then the cen­ter $\mathfrak{C}$ of $\mathfrak{U}$ is sta­bil­ized by $J$ and the re­stric­tion of $J$ to $\mathfrak{C}$ is either the iden­tity map or an auto­morph­ism of peri­od two. Ac­cord­ingly, the in­vol­u­tion is of first kind or second kind. Al­bert showed that $\mathfrak{U}$ has an in­vol­u­tion if and only if for any $m=1,2,\dots$, the mat­rix al­gebra $M_m(\mathfrak{U})$ has an in­vol­u­tion hav­ing the same ef­fect on the cen­ter (which can be iden­ti­fied with the cen­ter of the mat­rix al­gebra). He showed also that if $\mathfrak{U}$ has an in­vol­u­tion $J$ and $x$ is an ele­ment of $\mathfrak{U}$ whose min­im­um poly­no­mi­al over the cen­ter $\mathfrak{C}$ is ir­re­du­cible and has coef­fi­cients that are $J$-sym­met­ric, then $\mathfrak{U}$ has an in­vol­u­tion $T$ leav­ing $x$ fixed and hav­ing the same ef­fect on $\mathfrak{C}$ as $J$. As­sum­ing $\mathfrak{U}$ is of di­men­sion $n^2$ over its cen­ter $\mathfrak{C}$ and con­tains a sub­field of the form $\mathfrak{X}{\otimes }_{\mathfrak{F}}\mathfrak{C}$, where $\mathfrak{X}$ is $n$-di­men­sion­al Galois over $\mathfrak{F}$, and $\mathfrak{C}$ is either $\mathfrak{F}$ or a sep­ar­able quad­rat­ic ex­ten­sion of $\mathfrak{F}$, he gave a ne­ces­sary and suf­fi­cient con­di­tion in terms of a factor set for $\mathfrak{U}$ to have an in­vol­u­tion. This was used to give con­struc­tions which in prin­ciple yield all simple al­geb­ras with in­vol­u­tion. Al­bert also used these res­ults to prove that a cent­ral simple al­gebra has an in­vol­u­tion if and only if it has ex­po­nent one or two in the Brauer group. One can com­bine this with one of the res­ults of I to con­clude that a cent­ral di­vi­sion al­gebra of de­gree four has an in­vol­u­tion if and only if it is a tensor product of qua­ternion al­geb­ras. It seems un­likely that this is true for de­gree great­er than four but we be­lieve that this re­mains an open ques­tion.

Al­bert proved that if a di­vi­sion al­gebra over a num­ber field has an in­vol­u­tion of first kind, then the al­gebra is a qua­ternion al­gebra over its cen­ter. Moreover, he de­term­ined the di­vi­sion al­geb­ras over num­ber fields hav­ing in­vol­u­tions of second kind. He showed that any such al­gebra is cyc­lic $(\mathfrak{Z},s,\gamma )$ over its cen­ter $\mathfrak{C}$, where the cyc­lic field $\mathfrak{Z}$ over $\mathfrak{C}$ has the form ${\mathfrak{Z}}_0{\otimes }_{{\mathfrak{C}}_0}\mathfrak{C}$, ${\mathfrak{C}}_0$ the sub­field of sym­met­ric ele­ments of $\mathfrak{C}$, ${\mathfrak{Z}}_0$ cyc­lic over ${\mathfrak{C}}_0$, and $\mathfrak{C}$ sep­ar­able quad­rat­ic over ${\mathfrak{C}}_0$. Moreover, $\gamma \in \mathfrak{C}$, and if $\overline{\gamma }$ is its con­jug­ate in $\mathfrak{C}$ over ${\mathfrak{C}}_0$, then $\gamma \overline{\gamma }$ is a norm of an ele­ment of ${\mathfrak{Z}}_0$. Con­versely any di­vi­sion al­gebra hav­ing the in­dic­ated cyc­lic struc­ture does have an in­vol­u­tion of second kind. In a later pa­per [47], Al­bert ob­tained a sim­il­ar res­ult for di­vi­sion al­geb­ras of de­gree three (over their cen­ters) for ar­bit­rary base fields.

These res­ults, es­pe­cially those on di­vi­sion al­geb­ras with in­vol­u­tions over num­ber fields, provided the ma­chinery for the solu­tion of the prob­lem of mul­ti­plic­a­tion al­geb­ras for Riemann matrices. Us­ing Ros­ati’s the­or­em, one sees that the cen­ter $\mathfrak{C}$ of such an al­gebra is totally real if the (Ros­ati) in­vol­u­tion is of first kind and is a pure ima­gin­ary quad­rat­ic ex­ten­sion of a totally real field if the in­vol­u­tion is of second kind. Be­sides these con­di­tions on $\mathfrak{C}$, there are sup­ple­ment­ary con­di­tions on the qua­ternion di­vi­sion al­geb­ras and the cyc­lic al­geb­ras, which oc­curred in our de­scrip­tion of the di­vi­sion al­geb­ras with in­vol­u­tion over num­ber fields, that must be ful­filled for these to be mul­ti­plic­a­tion al­geb­ras of Riemann matrices. In Al­bert’s proof of the suf­fi­ciency of the con­di­tions he had de­rived, he made use of the Hil­bert ir­re­du­cib­il­ity the­or­em for num­ber fields.

In an ex­pos­i­tion [e30] of the the­ory of Riemann matrices, C. L. Siegel made some not­able im­prove­ments on Al­bert’s res­ults. We should men­tion also that Weyl in [e11] gave an al­tern­at­ive treat­ment of the sub­ject based on Brauer factor sets.

From about 1942 to 1965, when he re­turned to the prob­lem of ex­ist­ence of non­cyc­lic as­so­ci­at­ive di­vi­sion al­geb­ras of prime de­gree, most of Al­bert’s re­search was in the area of nonas­so­ci­at­ive al­gebra: struc­ture the­ory of nonas­so­ci­at­ive al­geb­ras, quasig­roups, nonas­so­ci­at­ive di­vi­sion rings, and nondesar­guesian pro­ject­ive planes. In our ac­count of his con­tri­bu­tions to this rather broad field of math­em­at­ics, we shall be se­lect­ive, pick­ing out what we con­sider his most im­port­ant work — judged from the cri­terion of gen­er­al math­em­at­ic­al in­terest. From this point of view, Al­bert’s dis­cov­er­ies on Jordan al­geb­ras are un­doubtedly his most im­port­ant ones in nonas­so­ci­at­ive al­gebra, and these are per­haps on a par with his work on as­so­ci­at­ive al­geb­ras and Riemann matrices. We shall be­gin our ac­count with this work, and we shall first sketch the story of Jordan al­geb­ras be­fore Al­bert took them up as a sub­ject of in­tens­ive study.

The study of the class of al­geb­ras which now bear his name was ini­ti­ated in 1932 by the phys­i­cist, P. Jordan. His de­clared ob­ject­ive was to achieve a bet­ter form­al­ism for quantum mech­an­ics than one based on sel­fad­joint op­er­at­ors in Hil­bert space. Ob­serving that the set of these op­er­at­ors is a vec­tor space over $\mathit{R}$ which is closed un­der the product $\mathit{A}\cdot \mathit{B}=\frac{1}{2}(AB+BA)$, where $AB$ is the usu­al as­so­ci­at­ive product, and that this sym­met­rized product is com­mut­at­ive and sat­is­fies the iden­tity $(A^2\cdot B)\cdot A=A^2\cdot (B\cdot A)$, he pro­posed to con­sider al­geb­ras in which the product com­pos­i­tion sat­is­fies these two con­di­tions. He en­lis­ted the help of von Neu­mann and Wign­er in his study, and as a res­ult of their col­lab­or­a­tion, there ap­peared, in 1934, a pa­per en­titled On an al­geb­ra­ic gen­er­al­iz­a­tion of the quantum mech­an­ic­al form­al­ism which was a gem in the struc­ture the­ory of al­geb­ras. In this pa­per [e9], Jordan, von Neu­mann and Wign­er ob­tained a com­plete de­term­in­a­tion of the fi­nite di­men­sion­al (nonas­so­ci­at­ive) al­geb­ras over $\mathit{R}$ sat­is­fy­ing the fol­low­ing con­di­tions: I. Form­al real­ity in the sense of Artin–Schreier, that is, the re­quire­ment that the only re­la­tions of the form $\sum a_i^2=0$ in the al­geb­ras are the trivi­al ones in which every $a_i=0$. II. Com­mut­ativ­ity of the product $(ab=ba)$ and the iden­tity $(a^{2}b)a=a^2(ba)$.

They showed that the al­geb­ras sat­is­fy­ing these con­di­tions are dir­ect sums of ideals that are simple al­geb­ras, and they de­term­ined the simple ones as be­long­ing to one of the fol­low­ing classes:

1. The vec­tor space over $\mathit{R}$ of $n\times n$ her­mitian matrices with entries in $\mathit{R}$, $\mathit{C}$, or Hamilton’s qua­ternion al­gebra $\mathit{H}$, en­dowed with the al­gebra struc­ture in which the product is $a\cdot b=\frac{1}{2}(ab+ba)$ in terms of the usu­al mat­rix product $ab$.

2. The al­geb­ras over $\mathit{R}$ with bases $(1,e_1,e_2,\dots ,e_n)$ and mul­ti­plic­a­tion defined by the table $e_i{e}_j={\delta }_{ij}1$, 1 the unit.

3. The al­gebra $M_3^8$ of $3\times 3$ her­mitian matrices with entries in the al­gebra $\mathit{O}$ of Cay­ley num­bers, en­dowed with the product $a\cdot b=\frac{1}{2}(ab+ba)$, where, as be­fore, $ab$ is the usu­al mat­rix product.

Con­versely, the al­geb­ras lis­ted sat­is­fy the above con­di­tions.

Now one defines a Jordan al­gebra over a field $\mathfrak{F}$ of char­ac­ter­ist­ic $\ne 2$ as an al­gebra in which the product $ab$ sat­is­fies $$ab=ba,(a^{2}b)a=a^2(ba).$$ Among these are in­cluded the spe­cial Jordan al­geb­ras which are iso­morph­ic to sub­spaces of an as­so­ci­at­ive al­gebra closed un­der the Jordan product $a\cdot b=\frac{1}{2}(ab+ba)$ and re­garded as al­geb­ras re­l­at­ive to this product. Evid­ently, the al­geb­ras in Jordan, von Neu­mann, Wign­er’s class (1) are spe­cial. It is easy to see, us­ing Clif­ford al­geb­ras, that this is the case also for al­geb­ras of their class (2). They con­jec­tured that $M_3^8$ is not spe­cial, and they pro­posed the proof of this as a prob­lem to Al­bert, who showed that this is in­deed the case: $M_3^8$ is an ex­cep­tion­al (= non­spe­cial) Jordan al­gebra [10].

After this brief en­counter with the Jordan the­ory, a num­ber of years elapsed be­fore Al­bert re­turned to the sub­ject. In a series of three pa­pers ap­pear­ing in 1946, 1947 and 1950, Al­bert [29], [31], [35] de­veloped the ba­sic struc­ture the­ory of fi­nite di­men­sion­al Jordan al­geb­ras over a field of char­ac­ter­ist­ic not two. Since it is in­ter­est­ing to ob­serve how the sub­ject evolved in Al­bert’s hands, we shall give a brief in­dic­a­tion of the con­tents of each of these pa­pers.

In the first one he con­sidered Jordan al­geb­ras defined con­cretely as Jordan al­geb­ras of lin­ear trans­form­a­tions of a fi­nite di­men­sion­al vec­tor space, that is, sub­spaces of $\mathrm{End}\mathfrak{B}$ closed un­der the Jordan product $A\cdot B$. He proved ana­logues for these al­geb­ras of Lie’s and En­gel’s the­or­ems on Lie al­geb­ras. As­sum­ing the base field is of char­ac­ter­ist­ic 0 (so trace ar­gu­ments can be used) he showed that if the al­geb­ras con­tain no nil ideals they are dir­ect sums of simple al­geb­ras. Moreover, he de­term­ined the simple al­geb­ras over an al­geb­ra­ic­ally closed field of char­ac­ter­ist­ic 0. This de­term­in­a­tion is quite sim­il­ar to that of Jordan, von Neu­mann and Wign­er’s of simple form­ally real Jordan al­geb­ras: Class (2) is un­changed, and the modi­fic­a­tion re­quired in the defin­i­tion of class (1) is that $\mathit{R}$, $\mathit{C}$ and $\mathit{H}$ be re­placed by the split com­pos­i­tion al­geb­ras of di­men­sions 1, 2 and 4 over the giv­en base fields. (These are the al­geb­ras which oc­cur in Hur­witz’s prob­lem on quad­rat­ic forms per­mit­ting com­pos­i­tion.) Nat­ur­ally there is no class (3) since the al­geb­ras un­der con­sid­er­a­tion are spe­cial by defin­i­tion. Ac­tu­ally Al­bert de­term­ined a more gen­er­al class of so-called re­duced al­geb­ras over an ar­bit­rary field of char­ac­ter­ist­ic 0. However, the res­ult he ob­tained in the al­geb­ra­ic­ally closed case is ad­equate to per­mit the de­term­in­a­tion of all the spe­cial simple al­geb­ras over an ar­bit­rary field us­ing the meth­od of des­cent. This was done by Kalisch [e18] and F. D. Jac­ob­son and N. Jac­ob­son [e20].

In his second pa­per, Al­bert dealt with ab­stract fi­nite di­men­sion­al Jordan al­geb­ras over any field of char­ac­ter­ist­ic not two. He showed that nil al­geb­ras of this type are nil­po­tent in the sense that there ex­ists an in­teger $r$ such that the product of any $r$ ele­ments of the al­gebra in any as­so­ci­ation is 0. He proved that if the char­ac­ter­ist­ic is 0 and $\mathfrak{U}$ has no nil ideals $\ne 0$, then $\mathfrak{U}$ is a dir­ect sum of simple al­geb­ras, and he de­term­ined the simple ones over an al­geb­ra­ic­ally closed field of char­ac­ter­ist­ic 0. Here, one does have ex­cep­tion­al al­geb­ras, and the only simple one over an al­geb­ra­ic­ally closed field of char­ac­ter­ist­ic 0 is the ana­logue of $M_3^8$ in which the clas­sic­al Cay­ley al­gebra is re­placed by a split Cay­ley al­gebra. We shall refer to this as the split ex­cep­tion­al simple Jordan al­gebra.

In his third pa­per, Al­bert ex­ten­ded these res­ults ex­cept for a small gap (which was filled by Jac­ob­son in [e26]) to the char­ac­ter­ist­ic $p\ne 2$ case.

A par­tic­u­larly in­ter­est­ing class of Jordan al­geb­ras is that of the fi­nite di­men­sion­al ex­cep­tion­al cent­ral simple Jordan al­geb­ras. If $\mathfrak{F}$ is the base field and $\overline{\mathfrak{F}}$ is its al­geb­ra­ic clos­ure, then $\mathfrak{U}$ is in this class if and only if the ex­ten­sion al­gebra ${\mathfrak{U}}^{\overline{\mathfrak{F}}}$ is the split ex­cep­tion­al simple Jordan al­gebra pre­vi­ously defined. The the­ory of these ex­cep­tion­al al­geb­ras is in­ter­twined with a num­ber of oth­er “ex­cep­tion­al” phe­nom­ena, not­ably, ex­cep­tion­al Lie groups and ex­cep­tion­al geo­met­ries (e.g. Cay­ley pro­ject­ive planes). Al­bert made a num­ber of im­port­ant con­tri­bu­tions to this the­ory, some­times in col­lab­or­a­tion with oth­ers. We have already noted that his first pa­per on Jordan al­geb­ras es­tab­lished the ex­cep­tion­al char­ac­ter of $M_3^8$. In 1959 Al­bert and Paige, in a joint pa­per [43], proved a much stronger res­ult: $M_3^8$ is not a ho­mo­morph­ic im­age of any spe­cial Jordan al­gebra. As a con­sequence of this and a res­ult of Cohn’s [e24], one can con­clude that the free Jordan al­gebra with three gen­er­at­ors is not spe­cial.

One can dis­tin­guish two types of ex­cep­tion­al simple Jordan al­geb­ras: the re­duced ones and the di­vi­sion al­geb­ras. The first con­tains idem­potents $\ne 0,1$, and in the second the sub­al­geb­ras gen­er­ated by single ele­ments and 1 are as­so­ci­at­ive fields. It can be shown that the re­duced ones have the form $\mathfrak{H}({\mathit{O}}_3,\gamma )$ the al­gebra of $3\times 3$ matrices $A$ with entries in some (gen­er­al­ized) Cay­ley al­gebra $\mathit{O}$, which are $\gamma$-her­mitian in the sense that ${\gamma }^{-1}{}^t\overline{A}\gamma =A$, where $\gamma$ is a 3-rowed di­ag­on­al mat­rix with entries in the base field, and $\overline{A}$ is ob­tained by re­pla­cing each Cay­ley num­ber entry $a_{ij}$ by its con­jug­ate ${\overline{a}}_{ij}$. The prob­lem of de­term­in­ing con­di­tions for the iso­morph­ism of two such al­geb­ras was stud­ied by Al­bert and Jac­ob­son [40]. It was shown in this pa­per that iso­morph­ism of the Jordan al­geb­ras im­plied iso­morph­ism of the Cay­ley al­geb­ras oc­cur­ring in their defin­i­tions, and they ob­tained some rather com­plic­ated sup­ple­ment­ary con­di­tions for iso­morph­ism. These suf­ficed to give a com­plete clas­si­fic­a­tion of re­duced ex­cep­tion­al simple Jordan al­geb­ras over num­ber fields. (See also [e29], [e35].)

The first con­struc­tion of ex­cep­tion­al Jordan di­vi­sion al­geb­ras is due to Al­bert [41], [49]. He showed also that no such al­geb­ras ex­ist over num­ber fields. On the oth­er hand, if $\mathfrak{F}$ is any field over which there ex­ist cent­ral as­so­ci­at­ive di­vi­sion al­geb­ras of de­gree three (e.g. a num­ber field), then there ex­ist ex­cep­tion­al Jordan di­vi­sion al­geb­ras over the field $\mathfrak{F}(t)$ ob­tained by ad­join­ing an in­de­term­in­ate $t$ to $\mathfrak{F}$. Al­bert used a meth­od of des­cent in his study of ex­cep­tion­al Jordan di­vi­sion al­geb­ras. Sub­sequently con­sid­er­ably sim­pler “ra­tion­al” con­struc­tions were giv­en by Tits ([e32], p. 412).

From the ab­stract point of view, a very nat­ur­al class of al­geb­ras (or rings) is the class sat­is­fy­ing the power as­so­ci­ativ­ity con­di­tion: sub­al­geb­ras (or sub­rings) gen­er­ated by single ele­ments are as­so­ci­at­ive. This in­cludes Jordan al­geb­ras, al­tern­at­ive al­geb­ras (defined by the iden­tit­ies $a^{2}b=a(ab)$ and $b{a}^2=(ba)a$), as­so­ci­at­ive al­geb­ras, and a num­ber of oth­er in­ter­est­ing types of al­geb­ras.

Al­bert ini­ti­ated the study of power as­so­ci­at­ive rings (without fi­nite­ness con­di­tions) in a pa­per pub­lished in [32]. The con­di­tions of power as­so­ci­ativ­ity are that for any $a$ one has the power for­mula $$a^m{a}^n=a^{m+n},m,n=1,2,3,\dots ,$$ where $a^m$ is defined in­duct­ively by $a^1=a$, $a^k=a^{k-1}a$. In [32] Al­bert showed that if the ad­dit­ive group of a nonas­so­ci­at­ive ring $\mathfrak{U}$ has no tor­sion, then $\mathfrak{U}$ is power as­so­ci­at­ive if and only if it sat­is­fies the two iden­tit­ies $a{a}^2=a^3$ and $(a^2{)}^2=a^4$. These res­ults were ob­tained by some clev­er in­duct­ive ar­gu­ments based on lin­ear­iz­a­tions of the as­sumed iden­tit­ies. These lin­ear­iz­a­tions and com­mut­ativ­ity yield also the cru­cial res­ult that in a com­mut­at­ive power as­so­ci­at­ive ring the map $e_R:x\to xe$, de­term­ined by an idem­potent $e$, sat­is­fies the quad­rat­ic equa­tion $(2e_R-1)(e_R-1)e_R=0$. If one as­sumes that the ad­dit­ive group ad­mits the op­er­at­or $\frac{1}{2}$, then one ob­tains the Peirce de­com­pos­i­tion re­l­at­ive to $$e:\mathfrak{U}={\mathfrak{U}}_0(e)\oplus {\mathfrak{U}}_{1/2}(e)\oplus {\mathfrak{U}}_1(e)\text{where}{\mathfrak{U}}_i(e)=\{x_i\mid x_{i}e=i{x}_i\}.$$ One also has ex­ten­sions of this to Peirce de­com­pos­i­tions re­l­at­ive to or­tho­gon­al idem­potents. In Al­bert’s hands, these Peirce de­com­pos­i­tions be­came power­ful tools for in­vest­ig­at­ing power as­so­ci­at­ive rings. He ob­tained a num­ber of strik­ing res­ults by this meth­od. We men­tion two:

Let $\mathfrak{U}$ be a simple com­mut­at­ive power as­so­ci­at­ive ring whose ad­dit­ive group con­tains no ele­ments of or­ders 2, 3 or 5 and ad­mits the op­er­at­or $\frac{1}{2}$. Sup­pose $\mathfrak{U}$ con­tains two nonzero or­tho­gon­al idem­potents $e$ and $f$ such that $e+f$ is not a unit. Then $\mathfrak{U}$ sat­is­fies the Jordan iden­tity $(a^{2}b)a=a^2(ba)$ [35].

Any simple al­tern­at­ive ring con­tain­ing an idem­potent $e\ne 0,1$ is either as­so­ci­at­ive or a Cay­ley al­gebra over its cen­ter [37].

The ul­ti­mate res­ult on simple al­tern­at­ive rings is due to Klein­feld [e23], [e27]. This states that all simple al­tern­at­ive rings are either as­so­ci­at­ive or Cay­ley al­geb­ras. Al­bert’s the­or­em was used as a step in the first proof of Klein­feld’s the­or­em.

In [36] and [41] Al­bert proved a gen­er­al­iz­a­tion for power as­so­ci­at­ive rings of Wed­der­burn’s cel­eb­rated the­or­em on the com­mut­ativ­ity of fi­nite as­so­ci­at­ive di­vi­sion rings. Call an al­gebra over a field strictly power as­so­ci­at­ive if all the al­geb­ras ob­tained by ex­tend­ing the base field are power as­so­ci­at­ive. Also one defines a (nonas­so­ci­at­ive) di­vi­sion ring by the prop­erty that the left and right mul­ti­plic­a­tions $x\to ax$ and $x\to xa$ are biject­ive for any $a\ne 0$ in the ring. Then Al­bert proved that any fi­nite strictly power as­so­ci­at­ive di­vi­sion al­gebra of char­ac­ter­ist­ic $\ne 2$ is as­so­ci­at­ive and com­mut­at­ive. Al­bert based his proof on the de­term­in­a­tion of the simple Jordan al­geb­ras over an ar­bit­rary field due to F. D. Jac­ob­son and N. Jac­ob­son [e20] and a res­ult of his own on ex­cep­tion­al Jordan al­geb­ras [41]. (Later Mc­Crim­mon gave al­tern­at­ive proofs which are in­de­pend­ent of the struc­ture the­ory [e31], [e34].) A num­ber of con­struc­tions of nonas­so­ci­at­ive and non­com­mut­at­ive di­vi­sion rings are due to Al­bert [46], [36], [45]. These yield ex­amples of nondesar­guesian pro­ject­ive planes in­clud­ing some fi­nite ones.

Al­bert had a hand in the dis­cov­ery of sev­er­al new classes of simple Lie al­geb­ras of prime char­ac­ter­ist­ic (see [e25], [38] and [e28]). Re­cently these res­ults have taken on ad­ded luster be­cause of the dis­cov­ery by Kostrikin and Sha­far­e­v­itch [e33] that these Lie al­geb­ras can be re­garded as char­ac­ter­ist­ic $p$ ver­sions of in­fin­ite di­men­sion­al Lie al­geb­ras which had oc­curred in Élie Cartan’s work on con­tact trans­form­a­tions.

Al­bert and his stu­dents and fol­low­ers also stud­ied a num­ber of oth­er classes of nonas­so­ci­at­ive al­geb­ras defined by iden­tit­ies. Un­til now the res­ults which have been ob­tained on these ap­pear to be of in­terest only to spe­cial­ists in the field. We shall there­fore re­frain from giv­ing any in­dic­a­tion of these res­ults. Al­bert wrote sev­er­al pa­pers on gen­er­al nonas­so­ci­at­ive the­ory. In one of these [25] he gave a defin­i­tion of a rad­ic­al for any fi­nite di­men­sion­al nonas­so­ci­at­ive al­gebra. Since the the­ory of the rad­ic­al is quite in­ter­est­ing and de­serves to be bet­ter known than it is at present, we take this op­por­tun­ity to sketch what we be­lieve is an im­proved ver­sion of this the­ory.

Let $\mathfrak{U}$ be a fi­nite di­men­sion­al nonas­so­ci­at­ive al­gebra over a field. Then $\mathfrak{U}$ is called simple if ${\mathfrak{U}}^2\ne 0$ and $\mathfrak{U}$ has no ideals $\ne 0$, $\mathfrak{U}$. $\mathfrak{U}$ is semisimple if it is a dir­ect sum of ideals which are simple al­geb­ras. Fol­low­ing the pat­tern of as­so­ci­at­ive ring the­ory, it is nat­ur­al to define the rad­ic­al $\mathrm{rad}\mathfrak{U}$ to be the in­ter­sec­tion of the set of ideals $\mathfrak{B}$ of $\mathfrak{U}$ such that $\mathfrak{U}/\mathfrak{B}$ is simple. This defin­i­tion im­plies that if no $\mathfrak{B}$’s, such that $\mathfrak{U}/\mathfrak{B}$ is simple, ex­ist then $\mathfrak{U}=\mathrm{rad}\mathfrak{U}$. In any case $\mathfrak{U}/\mathrm{rad}\mathfrak{U}$ is semisimple or 0, and $\mathrm{rad}\mathfrak{U}$ is con­tained in every ideal $\mathfrak{D}$ of $\mathfrak{U}$ such that $\mathfrak{U}/\mathfrak{D}$ is semisimple (see, for ex­ample, Jac­ob­son, Struc­ture of Rings, p. 41). This im­plies that $A\ne 0$ is semisimple if and only if $\mathrm{rad}\mathfrak{U}=0$.

One ob­tains im­port­ant in­form­a­tion on an al­gebra $\mathfrak{U}$ in look­ing at its mul­ti­plic­a­tion al­gebra $M(\mathfrak{U})$. This is the sub­al­gebra of the as­so­ci­at­ive al­gebra End $\mathfrak{U}$ of lin­ear trans­form­a­tions in $\mathfrak{U}$ gen­er­ated by 1 and the left and right mul­ti­plic­a­tions ($a_L:x\to ax$, $a_R:x\to xa$) of $\mathfrak{U}$. The cent­ral­izer of $M(\mathfrak{U})$ in End $\mathfrak{U}$ is called the centroid $C(\mathfrak{U})$ of $\mathfrak{U}$. The study of $M(\mathfrak{U})$ and $C(\mathfrak{U})$ was ini­ti­ated by Jac­ob­son  [e14] (see also Jac­ob­son, Lie Al­geb­ras, pp. 290–295). Al­bert’s res­ults on $\mathrm{rad}\mathfrak{U}$, as we shall show, amount to a for­mula for $\mathrm{rad}\mathfrak{U}$ in terms of $\mathrm{rad}M(\mathfrak{U})$. We shall call $\mathfrak{U}$ re­duct­ive if $\mathfrak{U}$ is a dir­ect sum of ideals which are simple al­geb­ras and the ideal $\mathfrak{Z}=\{z\mid \mathfrak{U}z=0=z\mathfrak{U}\}$. The ele­ments of $\mathfrak{Z}$ are called ab­so­lute zero di­visors. One can show that $\mathfrak{U}\ne 0$ is re­duct­ive if and only if $M(\mathfrak{U})$ is semisimple. Hence $\mathfrak{U}$ is semisimple if and only if $M(\mathfrak{U})$ is semisimple and 0 is the only ab­so­lute zero di­visor in $\mathfrak{U}$. Now let $\mathfrak{N}$ be the rad­ic­al of $M(\mathfrak{U})$ and $\mathfrak{R}$ the ideal in $\mathfrak{U}$ such that $\mathfrak{R}/\mathfrak{N}\mathfrak{U}$ is the ideal of ab­so­lute zero di­visors of $\mathfrak{U}/\mathfrak{N}\mathfrak{U}$. Then $\mathfrak{R}=\mathrm{rad}\mathfrak{U}$. Al­bert’s or­der of ideas in his pa­per on the rad­ic­al is the re­verse of what we have in­dic­ated; namely, he uses the ideal $\mathfrak{R}$ as his defin­i­tion of the rad­ic­al, then proves it has the two ba­sic prop­er­ties that $\mathfrak{U}/\mathfrak{R}$ is semisimple and $\mathfrak{R}$ is con­tained in every ideal $\mathfrak{D}$ such that $\mathfrak{U}/\mathfrak{D}$ is semisimple.

For cer­tain im­port­ant classes of al­geb­ras (e.g. as­so­ci­at­ive, al­tern­at­ive, Jordan), $\mathrm{rad}\mathfrak{U}$ co­in­cides with the max­im­al nil ideal. For Lie al­geb­ras of char­ac­ter­ist­ic 0, $\mathrm{rad}\mathfrak{U}$ is the max­im­al solv­able ideal. On the oth­er hand, Al­bert has giv­en an ex­ample of an al­gebra in which $\mathrm{rad}\mathfrak{U}$ is an as­so­ci­at­ive field. We could not res­ist re­cord­ing here a res­ult on the rad­ic­al which we have known for some time. This is a gen­er­al­iz­a­tion of a well-known the­or­em of Hoch­schild’s [e17] on de­riv­a­tions of as­so­ci­at­ive and Lie al­geb­ras.

This can be proved by us­ing the fact that the Lie al­gebra $\mathrm{Der}\mathfrak{U}$ of de­riv­a­tions is the Lie al­gebra of al­geb­ra­ic groups of auto­morph­isms of $\mathfrak{U}$. A more dir­ect proof, which is ap­plic­able also in some situ­ations in char­ac­ter­ist­ic $p\ne 0$, can be based on Al­bert’s defin­i­tion of $\mathrm{rad}\mathfrak{U}$. We ob­serve first that if $D$ is a de­riv­a­tion in $\mathfrak{U}$, then $$[D,a_L]\equiv D{a}_L-a_{L}D=(Da{)}_L\text{and}[D,a_R]=(Da{)}_R.$$ Hence $m\to [D,m]$ is a de­riv­a­tion in $M(\mathfrak{U})$ which we de­note as $\tilde{D}$. If $m\in M(\mathfrak{U})$ and $a\in \mathfrak{U}$, then $D(ma)=(\tilde{D}m)a+m(Da)$. It is eas­ily seen that $D$ sta­bil­izes $\mathrm{rad}\mathfrak{U}$ if $\tilde{D}$ sta­bil­izes $\mathrm{rad}M(\mathfrak{U})$. Our the­or­em then fol­lows from Hoch­schild’s the­or­em on as­so­ci­at­ive al­geb­ras.

In a pa­per [26] which ap­peared in 1942, Al­bert in­tro­duced a concept of iso­topy for nonas­so­ci­at­ive al­geb­ras. Let $\mathfrak{U}$ and $\mathfrak{B}$ be nonas­so­ci­at­ive al­geb­ras. Then $\mathfrak{U}$ and $\mathfrak{B}$ are called iso­topes if there ex­ist biject­ive lin­ear maps $P$ and $Q$ from $\mathfrak{B}$ to $\mathfrak{U}$ and a biject­ive lin­ear map $C$ from $\mathfrak{U}$ to $\mathfrak{B}$ such that for $x,y\in \mathfrak{B}$ we have $$xy=C((Px)(Qy)).$$ If $P=Q$ and $C=P^{-1}$, we have $P(xy)=(Px)(Py)$, so $P$ is an iso­morph­ism. Iso­topy is an equi­val­ence re­la­tion. If $\mathfrak{B}$ and $\mathfrak{U}$ are identic­al as sets, and $C=1$, we call $\mathfrak{B}$ a prin­cip­al iso­tope of $\mathfrak{U}$. Define a new mul­ti­plic­a­tion on $\mathfrak{U}$ by $u\circ v=(PCu)(QCv)$. This, along with the giv­en vec­tor space struc­ture, gives a new al­gebra which is a prin­cip­al iso­tope of $\mathfrak{U}$, and since $$xy=C((Px)(Qy))=C((PC{C}^{-1}x)(QC{C}^{-1}y))=C(C^{-1}x\circ C^{-1}y),$$ $\mathfrak{B}$ is iso­morph­ic to the prin­cip­al iso­tope defined by $\circ$. This re­duces the con­sid­er­a­tion to that of prin­cip­al iso­topes. Al­bert defined iso­topy also for quasig­roups [27], and he proved a num­ber of in­ter­est­ing res­ults on iso­topy of al­geb­ras and of quasig­roups. While these have not played an im­port­ant role in struc­ture the­ory, the concept of iso­topy has some im­port­ance in nondesar­guesian geo­metry (see [44]).

Al­bert wrote a num­ber of pa­pers [7], [13], [11], [14], [15] on the struc­ture of field ex­ten­sions. He was par­tic­u­larly in­ter­ested in ex­pli­cit con­struc­tions of cyc­lic field ex­ten­sions since these played an im­port­ant role in his in­vest­ig­a­tions of the struc­ture of di­vi­sion al­geb­ras. Al­bert’s res­ults on cyc­lic ex­ten­sions are presen­ted in a con­nec­ted fash­ion in Chapter IX of his al­gebra text Mod­ern High­er Al­gebra. There are nu­mer­ous ref­er­ences to these res­ults in Struc­ture of Al­geb­ras. For the case of de­gree $p^e$ and char­ac­ter­ist­ic $p$, one has an al­tern­at­ive meth­od due to Witt, based on Witt vec­tors, which provides a bet­ter sur­vey of cyc­lic and abeli­an ex­ten­sions (see for ex­ample, Jac­ob­son’s Lec­tures in Ab­stract Al­gebra, vol. III, pp. 124–140). On the oth­er hand, Al­bert’s res­ults on cyc­lic fields of de­gree $p^e$ and char­ac­ter­ist­ic $\ne p$ seem not to have been im­proved upon un­til now.

Al­bert was fas­cin­ated by the prob­lem of min­im­um num­ber of gen­er­at­ors for al­geb­ra­ic struc­tures. He proved [28] that any sep­ar­able as­so­ci­at­ive al­gebra is gen­er­ated by two ele­ments and, with John Thompson [42], proved that the pro­ject­ive un­im­od­u­lar group over a fi­nite field is gen­er­ated by two ele­ments, one of which has or­der two.

In a joint pa­per with Mucken­houpt [39], he proved that for any field $\mathfrak{F}$, any mat­rix of trace 0 in $M_n(\mathfrak{F})$ is an ad­dit­ive com­mut­at­or $[A,B]=AB-BA$. This sup­ple­men­ted an earli­er res­ult by Shoda [e15] for fields of char­ac­ter­ist­ic 0.

In [24] Al­bert proved that a fi­nite di­men­sion­al ordered di­vi­sion al­gebra is ne­ces­sar­ily com­mut­at­ive. This does not hold for in­fin­ite di­men­sion­al al­geb­ras, for Hil­bert has giv­en an ex­ample in the second edi­tion of his Grundla­gen der Geo­met­rie of a “twis­ted” power series di­vi­sion ring which is not com­mut­at­ive and which can be ordered. It is in­ter­est­ing to note that Hil­bert’s first at­tempt to give such an ex­ample in the first edi­tion of Grundla­gen can be seen to be wrong by in­vok­ing Al­bert’s the­or­em!

An­oth­er pretty res­ult of Al­bert’s gives a de­term­in­a­tion of the fi­nite di­men­sion­al ab­so­lute val­ued al­geb­ras over $\mathit{R}$. By this we mean a (nonas­so­ci­at­ive) al­gebra over $\mathit{R}$ which has a map $a\to |a|$ in­to $\mathit{R}$ with the usu­al prop­er­ties:

1. $|a|\geqq 0$ and $|a|=0$ if and only if $a=0$;

2. $|a+b|\leqq |a|+|b|$;

3. $|\alpha a|=|\alpha ||a|$ for $\alpha \in \mathit{R}$;

4. $|ab|=|a||b|$.

It had been con­jec­tured by Ka­plansky that if such an al­gebra has a unit, then it is al­tern­at­ive, and hence, by a clas­sic­al res­ult, it is ne­ces­sar­ily either $\mathit{R}$, $\mathit{C}$, Hamilton’s qua­ternion al­gebra $\mathit{H}$, or Cay­ley’s oc­to­nions $\mathit{O}$. Moreover, in all cases $|a|=|a\overline{a}{|}^{1/2}$, where $\overline{a}$ is the usu­al con­jug­ate. Al­bert proved this [30] and also showed that if all the con­di­tions ex­cept the ex­ist­ence of a unit hold, then the al­gebra is an iso­tope of $\mathit{R}$, $\mathit{C}$, $\mathit{H}$ or $\mathit{O}$. This res­ult was ex­ten­ded [33] to al­geb­ra­ic al­geb­ras over $\mathit{R}$ not as­sumed to be fi­nite di­men­sion­al.

Al­bert’s last pub­lished pa­per [53] — pub­lished posthum­ously — proves an in­ter­est­ing the­or­em on qua­ternion al­geb­ras: If ${\mathfrak{U}}_1$ and ${\mathfrak{U}}_2$ are two (gen­er­al­ized) qua­ternion di­vi­sion al­geb­ras over a field $\mathfrak{F}$ and ${\mathfrak{U}}_1{\otimes }_{\mathfrak{F}}{\mathfrak{U}}_2$ is not a di­vi­sion al­gebra, then ${\mathfrak{U}}_1$ and ${\mathfrak{U}}_2$ have a com­mon quad­rat­ic sub­field.

Our re­cit­al of Al­bert’s ma­jor achieve­ments gives no in­dic­a­tion of his meth­ods or, more broadly speak­ing, of his math­em­at­ic­al style, which was highly in­di­vidu­al­ist­ic. Per­haps its most char­ac­ter­ist­ic qual­it­ies were the dir­ect­ness of his ap­proach to a prob­lem and his power and stam­ina to stick with it un­til he achieved a com­plete solu­tion. He had a fant­ast­ic in­sight in­to what might be ac­com­plished by in­tric­ate and subtle cal­cu­la­tions of a highly ori­gin­al char­ac­ter. At times he could have ob­tained sim­pler proofs by us­ing more soph­ist­ic­ated tools (e.g. rep­res­ent­a­tion the­ory), and one can al­most al­ways im­prove upon his ar­gu­ments. However, this is of sec­ond­ary im­port­ance com­pared to the first break­through which es­tab­lishes a defin­it­ive res­ult. It was in this that Al­bert really ex­celled. He re­garded him­self as a “pure” al­geb­ra­ist and in a sense he was. However, his best work — the solu­tion of the prob­lem of mul­ti­plic­a­tion al­geb­ras of Riemann matrices — had its ori­gin in an­oth­er branch of math­em­at­ics. Moreover, he could ex­ploit ana­lyt­ic and num­ber the­or­et­ic res­ults when he needed them — as he did in this in­stance.