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Celebratio Mathematica

Leonard Eugene Dickson

Algebra  ·  U Chicago

Leonard Eugene Dickson 1874–1954

by A. A. Albert

Le­onard Eu­gene Dick­son was born in In­de­pend­ence, Iowa, on Janu­ary 22, 1874. He was a bril­liant un­der­gradu­ate at the Uni­versity of Texas, re­ceiv­ing his B.S. de­gree as va­le­dictori­an of his class in 1893. He was a chem­ist with the Texas Bio­lo­gic­al Sur­vey from 1892 to 1893. He served as a teach­ing fel­low at the Uni­versity of Texas, re­ceiv­ing the M.A. de­gree in 1894. He held a fel­low­ship at the Uni­versity of Chica­go from 1894 to 1896, and was awar­ded its first Ph.D. in math­em­at­ics in 1896. He spent the year 1896–1897 in Leipzig and Par­is, was in­struct­or in math­em­at­ics at the Uni­versity of Cali­for­nia 1897–1899, as­so­ci­ate pro­fess­or at Texas 1899–1900, as­sist­ant pro­fess­or at Chica­go 1900–1907, as­so­ci­ate pro­fess­or 1907–1910, and pro­fess­or in 1910. He was ap­poin­ted to the Eliakim Hast­ings Moore Dis­tin­guished Pro­fess­or­ship in 1928, and be­came pro­fess­or emer­it­us in 1939. He served as vis­it­ing pro­fess­or at the Uni­versity of Cali­for­nia in 1914, 1918, and 1922.

Pro­fess­or Dick­son was awar­ded the $1,000 A.A.A.S. Prize in 1924 for his work on the arith­met­ics of al­geb­ras. He was awar­ded the Cole Prize of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1928 for his book Al­be­gren und ihre Zah­len­the­or­ie [42]. He served as ed­it­or of the Monthly 1902–1908, and the Trans­ac­tions from 1911 to 1916, and he was pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety from 1916–1918. He was elec­ted to mem­ber­ship in the Na­tion­al Academy of Sci­ences in 1913, and was a mem­ber of the Amer­ic­an Philo­soph­ic­al So­ci­ety, the Amer­ic­an Academy of Arts and Sci­ences, and the Lon­don Math­em­at­ic­al So­ci­ety. He was also a for­eign mem­ber of the Academy of the In­sti­tute of France, and an hon­or­ary mem­ber of the Czechoslov­aki­an Uni­on of Math­em­at­ics and Phys­ics. He was awar­ded the hon­or­ary Sc.D. de­gree by Har­vard in 1936 and Prin­ceton in 1941.

Pro­fess­or Dick­son died in Texas on Janu­ary 17, 1954.

Dick­son was one of our most pro­lif­ic math­em­aticians. His bib­li­o­graphy (pre­pared by Mr. Richard Block, a stu­dent at the Uni­versity of Chica­go) con­tains 285 titles. Of these, eight­een are books, one a joint book with Miller and Blich­feldt [33]. One of the books is his ma­jor three-volume His­tory of the the­ory of num­bers [35], [36], [39], which would be a life’s work by it­self for a more or­din­ary man.

Dick­son was an in­spir­ing teach­er. He su­per­vised the doc­tor­ate dis­ser­ta­tions of at least fifty-five Chica­go Ph.D.s. He helped his stu­dents to get star­ted in re­search after the Ph.D. and his books had a world-wide in­flu­ence in stim­u­lat­ing re­search.

We now pass on to a brief dis­cus­sion of Dick­son’s re­search.

1. Linear groups

Dick­son’s first ma­jor re­search ef­fort was a study of fi­nite lin­ear groups. All but sev­en of his first forty-three pa­pers were on that sub­ject, and this por­tion of his work led to his fam­ous first book [3]. The lin­ear groups which had been in­vest­ig­ated by Galois, Jordan, and Ser­ret were all groups over the fields of \( p \) ele­ments. Dick­son gen­er­al­ized their res­ults to lin­ear groups over an ar­bit­rary fi­nite field. He ob­tained many new sys­tems of simple groups, and he closed his book with a still valu­able sum­mary of the known sys­tems of simple groups.

Dick­son’s work on lin­ear groups con­tin­ued un­til 1908, and he wrote about forty-four ad­di­tion­al pa­pers on the sub­ject. In these later pa­pers he stud­ied the iso­morph­ism of cer­tain simple groups and ques­tions about the ex­ist­ence of cer­tain types of sub­groups. He also de­rived a num­ber of the­or­ems on in­fin­ite lin­ear groups.

2. Finite fields and Chevalley’s theorem

In [3]. Dick­son gave the first ex­tens­ive ex­pos­i­tion of the the­ory of fi­nite fields. He ap­plied his deep know­ledge of that sub­ject not only to lin­ear groups but to oth­er prob­lems which we shall dis­cuss later. He stud­ied ir­re­du­cib­il­ity ques­tions over a fi­nite field in [11], the Galois the­ory in [10], and forms whose val­ues are squares in [19]. His know­ledge of the role of the non-null form was shown in [20]. In [16] Dick­son made the fol­low­ing state­ment: “For a fi­nite field it seems to be true that every form of de­gree \( m \) in \( m + 1 \) vari­ables van­ishes for val­ues not all zero in the field.” This res­ult was first proved by C. Che­val­ley in his pa­per Demon­stra­tion d’une hy­po­these de M. Artin, Hamb. Abh, vol. 11 (1935), pp. 73–75. At least the con­jec­ture should have been at­trib­uted to Dick­son, who ac­tu­ally proved the the­or­em for \( m = 2, 3 \).

3. Invariants

Sev­er­al of Dick­son’s early pa­pers were con­cerned with the prob­lems of the al­geb­ra­ic geo­metry of his time. For ex­ample, see [1], [5], [4]. This work led nat­ur­ally to his study of al­geb­ra­ic in­vari­ants, and his in­terest in fi­nite fields to mod­u­lar in­vari­ants. He wrote a ba­sic pa­per on the lat­ter sub­ject in [18], and many oth­er pa­pers on the sub­ject. In these pa­pers he de­voted a great deal of space to the de­tails of a num­ber of spe­cial cases. His book [24], on the clas­sic­al the­ory of al­geb­ra­ic in­vari­ants, was pub­lished in 1914, the year after the ap­pear­ance of his col­loqui­um lec­tures. His amaz­ing pro­ductiv­ity is at­tested to by the fact that he also pub­lished his book [25] on lin­ear al­geb­ras in 1914.

4. Algebras

Dick­son played a ma­jor role in re­search on lin­ear al­geb­ras. He began with a study of fi­nite di­vi­sion al­geb­ras in [6], [7], [9] and [8]. In these pa­pers he de­term­ined all three- and four-di­men­sion­al (nonas­so­ci­at­ive) di­vi­sion al­geb­ras over a field of char­ac­ter­ist­ic not two, a set of al­geb­ras of di­men­sion six, and a meth­od for con­struct­ing al­geb­ras of di­men­sion \( mk \) with a sub­field of the di­men­sion \( m \). In [15] he re­lated the the­ory of tern­ary cu­bic forms to the the­ory of three-di­men­sion­al di­vi­sion al­geb­ras. His last pa­per on non-as­so­ci­at­ive al­geb­ras [47] ap­peared in 1937 and con­tained ba­sic res­ults on al­geb­ras of de­gree two.

Ref­er­ence has already been made to Dick­son’s first book on lin­ear al­geb­ras. In that text he gave a proof of his res­ult that a real Cay­ley di­vi­sion al­gebra is ac­tu­ally a di­vi­sion al­gebra. He presen­ted the Cartan the­ory of lin­ear as­so­ci­at­ive al­geb­ras rather than the Wed­der­burn the­ory, but stated the res­ults of the lat­ter the­ory in his clos­ing chapter without proofs. The present value of this book is en­hanced by nu­mer­ous bib­li­o­graph­ic­al ref­er­ences.

Dick­son defined cyc­lic al­geb­ras in a Bul­let­in ab­stract of vol. 12 (1905–1906). His pa­per [21] on the sub­ject did not ap­pear un­til 1912, where he presen­ted a study of al­geb­ras of di­men­sion 16.

Dick­son’s work on the arith­met­ics of al­geb­ras first ap­peared in [37]. His ma­jor work on the sub­ject of arith­met­ics was presen­ted in [38] where he also gave an ex­pos­i­tion of the Wed­der­burn the­ory. See also [44] and [45].

The text [42] is a Ger­man ver­sion of [38]. However, the new ver­sion also con­tains the res­ults on crossed product al­geb­ras which had been pub­lished in [46], and con­tains many oth­er items of im­port­ance.

5. Theory of numbers

Dick­son al­ways said that math­em­at­ics is the queen of the sci­ences, and that the the­ory of num­bers is the queen of math­em­at­ics. He also stated that he had al­ways wished to work in the the­ory of num­bers and that he wrote his mo­nu­ment­al three-volume His­tory of the the­ory of num­bers [35], [36], [39] so that he could know all of the work which had been done in the sub­ject. His first pa­per [2] con­tained a gen­er­al­iz­a­tion of the ele­ment­ary Fer­mat the­or­em which arose in con­nec­tion with fi­nite-field the­ory. He was in­ter­ested in the ex­ist­ence of per­fect num­bers, and wrote [23] and [22] on the re­lated top­ic of abund­ant num­bers. His in­terest in Fer­mat’s last the­or­em ap­pears in [34], [12], [14], [13] and [17]. Dur­ing 1926–1930 he spent most of his en­ergy on re­search in the arith­met­ic the­ory of quad­rat­ic forms, in par­tic­u­lar on uni­ver­sal forms.

Dick­son’s in­terest in ad­dit­ive num­ber the­ory began in 1927 with [43]. He wrote many pa­pers on the sub­ject dur­ing the re­mainder of his life. The ana­lyt­ic res­ults of Vino­gradov gave Dick­son the hope of prov­ing the so-called ideal War­ing the­or­em. This he did in a long series of pa­pers. His fi­nal res­ult is an al­most com­plete veri­fic­a­tion of the con­jec­ture made by J. A. Euler in 1772. That con­jec­ture stated that every pos­it­ive in­teger is a sum of \( J \) \( n \)-th powers, where we write \( 3^n = 2^n q + r \), \( \,J = 2^n > r > 0 \), and \( J = 2^n + q - 2 \). Dick­son showed that if \( n > 6 \) this value is cor­rect un­less \( q+ r+3 > 2^n \). It is still not known wheth­er or not this last in­equal­ity is pos­sible, but if it does oc­cur the num­ber \( g(n) \) of such \( n \)-th powers re­quired to rep­res­ent all in­tegers is \( J + f \) or \( J+f-1 \), ac­cord­ing as \( fq + f+ q = 2^n \) or \( fq + f+q > 2^n \), where \( f \) is the greatest in­teger in \( (4/3)^n \) .

6. Miscellaneous

We close by men­tion­ing Dick­son’s in­terest in the the­ory of matrices which is best il­lus­trated by his text, Mod­ern al­geb­ra­ic the­or­ies [41]. His geo­met­ric work in [26], [30], [29], [32], [27], [31] and [28] must also be men­tioned, as well as his in­ter­est­ing mono­graph [40] on dif­fer­en­tial equa­tions from the Lie-group stand­point.

Works

[1]L. E. Dick­son: “A quad­rat­ic Cre­mona trans­form­a­tion defined by a con­ic,” Rend. Circ. Mat. Palermo 9 : 1 (1895), pp. 256–​259. Also pub­lished in Amer. Math. Monthly 2:7–8 (1895). JFM 26.​0610.​01 article

[2]L. E. Dick­son: “A gen­er­al­iz­a­tion of Fer­mat’s the­or­em,” Ann. Math. (2) 1 : 1–​4 (1899–1900), pp. 31–​36. French trans­la­tion pub­lished in C. R. Acad. Sci. Par­is 128 (1899). MR 1502247 JFM 30.​0185.​01 article

[3]L. E. Dick­son: Lin­ear groups with an ex­pos­i­tion of the Galois field the­ory. Sammlung von Lehr­büch­ern VI. B. G. Teub­n­er (Leipzig and Ber­lin), 1901. Re­pub­lished in 1958. JFM 32.​0128.​01 book

[4]L. E. Dick­son: “A class of groups in an ar­bit­rary realm con­nec­ted with the con­fig­ur­a­tion of the \( {}27 \) lines on a cu­bic sur­face,” Quart. J. Pure Ap­pl. Math. 33 (1901), pp. 145–​173. JFM 32.​0133.​01 article

[5]L. E. Dick­son: “The con­fig­ur­a­tions of the \( {}27 \) lines on a cu­bic sur­face and the \( {}28 \) bit­an­gents to a quart­ic curve,” Bull. Am. Math. Soc. 8 : 2 (1901), pp. 63–​70. MR 1557844 JFM 32.​0492.​01 article

[6]L. E. Dick­son: “On fi­nite al­geb­ras,” Na­chr. Ges. Wiss. Göt­tin­gen (1905), pp. 358–​393. JFM 36.​0138.​03 article

[7]L. E. Dick­son: “Lin­ear al­geb­ras in which di­vi­sion is al­ways uniquely pos­sible,” Trans. Am. Math. Soc. 7 : 3 (1906), pp. 370–​390. MR 1500755 JFM 37.​0111.​06 article

[8]L. E. Dick­son: “On com­mut­at­ive lin­ear al­geb­ras in which di­vi­sion is al­ways uniquely pos­sible,” Trans. Am. Math. Soc. 7 : 4 (1906), pp. 514–​522. MR 1500764 JFM 37.​0112.​01 article

[9]L. E. Dick­son: “On lin­ear al­geb­ras,” Amer. Math. Mon. 13 : 11 (November 1906), pp. 201–​205. MR 1516696 JFM 37.​0115.​03 article

[10]L. E. Dick­son: “On the the­ory of equa­tions in a mod­u­lar field,” Bull. Am. Math. Soc. 13 : 1 (1906), pp. 8–​10. MR 1558391 JFM 37.​0173.​02 article

[11]L. E. Dick­son: “Cri­ter­ia for the ir­re­du­cib­il­ity of func­tions in a fi­nite field,” Bull. Am. Math. Soc. 13 : 1 (1906), pp. 1–​8. MR 1558390 JFM 37.​0094.​01 article

[12]L. E. Dick­son: “On the last the­or­em of Fer­mat,” Mes­sen­ger of Math­em­at­ics 38 (1908), pp. 14–​32. Part II pub­lished in Quart. J. Pure Ap­pl. Math. 40 (1908). JFM 39.​0260.​01 article

[13]L. E. Dick­son: “On the con­gru­ence \( x^n+y^n+z^n=0 \) (mod \( p \)),” J. Reine An­gew. Math. 135 (1908), pp. 134–​141. JFM 39.​0260.​02 article

[14]L. E. Dick­son: “On the last the­or­em of Fer­mat, II,” Quart. J. Pure Ap­pl. Math. 40 (1908), pp. 27–​45. Part I pub­lished in Mes­sen­ger of Math­em­at­ics 38 (1908). JFM 39.​0260.​03 article

[15]L. E. Dick­son: “On triple al­geb­ras and tern­ary cu­bic forms,” Bull. Am. Math. Soc. 14 : 4 (1908), pp. 160–​169. MR 1558578 JFM 39.​0138.​03 article

[16]L. E. Dick­son: “On the rep­res­ent­a­tion of num­bers by mod­u­lar forms,” Bull. Am. Math. Soc. 15 : 7 (1909), pp. 338–​347. MR 1558771 JFM 40.​0269.​01 article

[17]L. E. Dick­son: “Lower lim­it for the num­ber of sets of solu­tions of \( x^e+y^e+z^e \equiv 0 \) (mod \( p \)),” J. Reine An­gew. Math. 135 (1909), pp. 181–​188. JFM 40.​0254.​04 article

[18]L. E. Dick­son: “Gen­er­al the­ory of mod­u­lar in­vari­ants,” Trans. Am. Math. Soc. 10 : 2 (1909), pp. 123–​158. MR 1500831 JFM 40.​0158.​01 article

[19]L. E. Dick­son: “Def­in­ite forms in a fi­nite field,” Trans. Am. Math. Soc. 10 : 1 (1909), pp. 109–​122. MR 1500830 JFM 40.​0268.​03 article

[20]L. E. Dick­son: “On non-van­ish­ing forms,” Quart. J. Pure Ap­pl. Math. 42 (1911), pp. 162–​171. JFM 42.​0138.​01 article

[21]L. E. Dick­son: “Lin­ear al­geb­ras,” Trans. Am. Math. Soc. 13 : 1 (1912), pp. 59–​73. MR 1500905 JFM 43.​0162.​09 article

[22]L. E. Dick­son: “Even abund­ant num­bers,” Amer. J. Math. 35 : 4 (October 1913), pp. 423–​426. MR 1506195 JFM 44.​0221.​01 article

[23]L. E. Dick­son: “Fi­nite­ness of the odd per­fect and prim­it­ive abund­ant num­bers with \( n \) dis­tinct prime factors,” Bull. Amer. Math. Soc. 19 : 6 (1913), pp. 285. Ab­stract for art­icle in Amer. J. Math. 35:4 (1913). JFM 44.​0220.​02 article

[24]L. E. Dick­son: Al­geb­ra­ic in­vari­ants. Math­em­at­ic­al mono­graphs 14. J. Wiley & Sons (New York), 1914. JFM 45.​0196.​10 book

[25]L. E. Dick­son: Lin­ear al­geb­ras. Cam­bridge Tracts in Math­em­at­ics and Math­em­at­ic­al Phys­ics 16. Cam­bridge Uni­versity Press, 1914. Re­pub­lished in 1960. JFM 45.​0189.​01 book

[26]L. E. Dick­son: “Pro­ject­ive clas­si­fic­a­tion of cu­bic sur­faces mod­ulo \( {}2 \),” Ann. Math. (2) 16 : 1–​4 (1914–1915), pp. 139–​157. MR 1502501 JFM 45.​0212.​01 article

[27]L. E. Dick­son: “The straight lines on mod­u­lar cu­bic sur­faces,” Proc. Nat. Acad. Sci. U.S.A. 1 : 4 (April 1915), pp. 248–​253. JFM 45.​0212.​02 article

[28]L. E. Dick­son: “In­vari­antive clas­si­fic­a­tion of pairs of con­ics mod­ulo \( {}2 \),” Amer. J. Math. 37 : 4 (October 1915), pp. 355–​358. An ab­stract was pub­lished as Bull. Am. Math. Soc. 22:1 (1915). MR 1506263 JFM 45.​0210.​02 article

[29]L. E. Dick­son: “Quart­ic curves mod­ulo \( {}2 \),” Trans. Am. Math. Soc. 16 : 2 (April 1915), pp. 111–​120. MR 1501003 JFM 45.​0211.​02 article

[30]L. E. Dick­son: “In­vari­antive the­ory of plane cu­bic curves mod­ulo \( {}2 \),” Amer. J. Math. 37 : 2 (April 1915), pp. 107–​116. MR 1506248 JFM 45.​0210.​03 article

[31]L. E. Dick­son: “Geo­met­ric­al and in­vari­antive the­ory of quart­ic curves mod­ulo \( {}2 \),” Amer. J. Math. 37 : 4 (October 1915), pp. 337–​354. MR 1507897 JFM 45.​0211.​01 article

[32]L. E. Dick­son: “Clas­si­fic­a­tion of quart­ic curves, mod­ulo \( {}2 \),” Mes­sen­ger of Math­em­at­ics 44 (1915), pp. 189–​192. JFM 45.​1235.​03 article

[33]G. A. Miller, H. F. Blich­feldt, and L. E. Dick­son: The­ory and ap­plic­a­tions of fi­nite groups. J. Wiley & Sons (New York), 1916. Re­pub­lished in 1938 and in 1961. JFM 46.​0171.​02 book

[34]L. E. Dick­son: “Fer­mat’s last the­or­em and the ori­gin and nature of the the­ory of al­geb­ra­ic num­bers,” Ann. Math. (2) 18 : 4 (1917), pp. 161–​187. MR 1503597 JFM 46.​0268.​02 article

[35]L. E. Dick­son: His­tory of the the­ory of num­bers, vol. I: Di­vis­ib­il­ity and prim­al­ity. Carne­gie In­sti­tu­tion (Wash­ing­ton, DC), 1919. See also Volume II and Volume III. Chelsea re­pub­lished in 1966. The whole series was re­pub­lished in 1934. JFM 47.​0100.​04 book

[36]L. E. Dick­son: His­tory of the the­ory of num­bers, vol. II: Di­o­phant­ine ana­lys­is. Carne­gie In­sti­tu­tion (Wash­ing­ton, DC), 1920. See also Volume I and \xlink{Volume III|}. Chelsea re­pub­lished in 1966. The whole series was re­pub­lished in 1934. book

[37]L. E. Dick­son: “Arith­met­ic of qua­ternions,” Bull. Am. Math. Soc. 27 : 7 (1921), pp. 300. Ab­stract only. Ab­stract for art­icle pub­lished in Proc. Lon­don Math. Soc. 20:1 (1922). JFM 48.​0130.​06 article

[38]L. E. Dick­son: Al­geb­ras and their arith­met­ics. Uni­versity of Chica­go Press, 1923. Re­pub­lished in 1938 and 1960. Ger­man trans­la­tion pub­lished as Al­gebren und ihre Zah­len­the­or­ie (1927). JFM 49.​0079.​01 book

[39]L. E. Dick­son: His­tory of the the­ory of num­bers, vol. III: Quad­rat­ic and high­er forms. Carne­gie In­sti­tu­tion (Wash­ing­ton, DC), 1923. With a chapter on the class num­ber by G. H. Cresse. See also Volume I and Volume II. Chelsea re­prin­ted in 1966. The whole series was re­pub­lished in 1934. JFM 49.​0100.​12 book

[40]L. E. Dick­son: “Dif­fer­en­tial equa­tions from the group stand­point,” Ann. Math. (2) 25 : 4 (1924), pp. 287–​378. MR 1502670 article

[41]L. E. Dick­son: Mod­ern al­geb­ra­ic the­or­ies. B. H. San­born & Co. (Chica­go), 1926. Re­pub­lished as Al­geb­ra­ic the­or­ies (1959). JFM 52.​0094.​01 book

[42]L. E. Dick­son: Al­gebren und ihre Zah­len­the­or­ie [Al­geb­ras and their arith­met­ics]. Ver­öf­fent­lichun­gen der Sch­weizerischen Math­em­at­ischen Gesell­schaft 4. Orell Füss­li (Zürich), 1927. Trans­la­tion of com­pletely re­vised and ex­ten­ded manuscript, with con­tri­bu­tion on ideal the­ory from An­dreas Speiser. Ger­man trans­la­tion of Al­geb­ras and their arith­met­ics (1923). JFM 53.​0112.​01 book

[43]L. E. Dick­son: “Ex­ten­sions of War­ing’s the­or­em on nine cubes,” Amer. Math. Mon. 34 : 4 (April 1927), pp. 177–​183. MR 1521139 JFM 53.​0134.​02 article

[44]L. E. Dick­son: “Out­line of the the­ory to date of the arith­met­ics of al­geb­ras,” pp. 95–​102 in Pro­ceed­ings of the In­ter­na­tion­al Math­em­at­ic­al Con­gress, 1924 (Toronto, 11–16 Au­gust, 1924), vol. 1. Edi­ted by J. C. Fields. Uni­versity of Toronto Press, 1928. JFM 54.​0160.​03 incollection

[45]L. E. Dick­son: “Fur­ther de­vel­op­ment of the the­ory of arith­met­ics of al­geb­ras,” pp. 173–​184 in Pro­ceed­ings of the In­ter­na­tion­al Math­em­at­ic­al Con­gress, 1924 (Toronto, 11–16 Au­gust, 1924), vol. 1. Edi­ted by J. C. Fields. Uni­versity of Toronto Press, 1928. JFM 54.​0161.​01 incollection

[46]L. E. Dick­son: “New di­vi­sion al­geb­ras,” Bull. Am. Math. Soc. 34 : 5 (1928), pp. 555–​560. See also Trans. Am. Math. Soc. 28:2 (1926) and C. R. Acad. Sci. Par­is 181 (1925). MR 1561617 JFM 54.​0161.​03 article

[47]L. E. Dick­son: “Lin­ear al­geb­ras with as­so­ci­ativ­ity not as­sumed,” Duke Math. J. 1 : 2 (1935), pp. 113–​125. MR 1545870 JFM 61.​0125.​01 Zbl 0012.​14801 article