Celebratio Mathematica

Leonard Eugene Dickson

Role modeling in mathematics:
The case of Leonard Eugene Dickson (1874–1954)

by Della Dumbaugh Fenster


In their book, The Emer­gence of the Amer­ic­an Math­em­at­ic­al Re­search Com­munity (1876–1900): J. J. Sylvester, Fe­lix Klein, and E. H. Moore, Kar­en Par­shall and Dav­id Rowe sug­gest the no­tion of peri­od­iz­a­tion (as op­posed to con­tinu­ity) as a means of his­tor­ic­ally in­vest­ig­at­ing math­em­at­ics [e53]. They char­ac­ter­ize four de­vel­op­ment­al peri­ods as fol­lows:

  • 1776–1876: Math­em­at­ics in the gen­er­al con­text of sci­entif­ic de­vel­op­ment

  • 1876–1900: Emer­gence of a re­search com­munity

  • 1900–1933: Con­sol­id­a­tion and growth of re­search tra­di­tions and in­sti­tu­tions

  • 1933–1960: In­flux of European math­em­aticians; Gov­ern­ment fund­ing ([e53], pp. 427–428).

With their view that the open­ing three dec­ades of this cen­tury wit­nessed a con­sol­id­a­tion and growth of Amer­ic­an re­search tra­di­tions and in­sti­tu­tions in math­em­at­ics, Par­shall and Rowe have joined the bor­der pieces of a large in­ter­lock­ing jig­saw puzzle. Their work chal­lenges his­tor­i­ans of Amer­ic­an math­em­at­ics to de­term­ine the pieces which make up the re­mainder of the pic­ture. This in­ter­est­ing and mul­ti­fa­ceted pro­ject un­folds as we ad­dress ques­tions such as: What re­search tra­di­tions came to­geth­er in this time peri­od? What math­em­aticians brought co­he­sion to these tra­di­tions and how did they do it? What in­sti­tu­tions emerged as re­search power­houses in math­em­at­ics and why? As these ques­tions have re­ceived at­ten­tion, they have quite nat­ur­ally brought in­to sharp fo­cus oth­er un­ex­pec­ted factors which con­trib­uted to the de­vel­op­ment of Amer­ic­an math­em­at­ics early in this cen­tury. The pro­cesses of ment­or­ing and role mod­el­ing, in par­tic­u­lar, rep­res­ent two such factors and, as a glance at the math­em­at­ic­al life of Le­onard Eu­gene Dick­son will re­veal, they proved to be among the more suc­cess­ful av­en­ues for the ad­vance­ment of math­em­at­ics in the United States.

By pur­su­ing both his doc­tor­al stud­ies and the ma­jor­ity of his ca­reer at the Uni­versity of Chica­go, Dick­son helped define an al­geb­ra­ic tra­di­tion at that in­sti­tu­tion. His math­em­at­ic­al con­tri­bu­tions — es­pe­cially his em­phas­is on the con­struc­tion of far-reach­ing the­or­ies on firm math­em­at­ic­al found­a­tions1  — brought con­sol­id­a­tion and growth to this al­geb­ra­ic en­deavor. Moreover, Dick­son fur­ther ad­vanced the Chica­go al­geb­ra­ic school by im­part­ing his stand­ards for sol­id math­em­at­ics to the next gen­er­a­tion of as­pir­ing doc­tor­al stu­dents.

Dick­son’s strong al­geb­ra­ic re­search pro­gram, situ­ated with­in the far-sighted Chica­go math­em­at­ics de­part­ment, ful­filled one of the broad­er con­cerns of Amer­ic­an math­em­at­ics, namely, the ef­fect­ive train­ing of fu­ture re­search­ers ([e53], p. 429). From its in­cep­tion, the Uni­versity of Chica­go provided a con­du­cive en­vir­on­ment for fu­ture math­em­aticians to learn not only math­em­at­ics but also those ex­tramathem­at­ic­al qual­it­ies of im­port for a ca­reer as a pro­duct­ive re­search­er. A list of the first gen­er­a­tion of gradu­ate stu­dents at Chica­go — Os­wald Veblen, Gil­bert A. Bliss, Le­onard Dick­son, and George D. Birk­hoff, to name only four — con­firms the in­sti­tu­tion’s suc­cess in this re­gard. This pa­per high­lights the con­tinu­ation of this suc­cess re­l­at­ive to the second gen­er­a­tion of Chica­go-trained re­search­ers.

Since Dick­son ac­quired many of the ideas and ideals he would pass on to the next gen­er­a­tion of as­pir­ing math­em­aticians from his gradu­ate stu­dent days at the then very young, but very vis­ion­ary, Uni­versity of Chica­go, we ini­tially con­sider what math­em­at­ic­al and ex­tramathem­at­ic­al at­trib­utes Dick­son ac­quired while purs­ing his doc­tor­ate. In our ef­forts to come to terms with the scope of Dick­son’s in­flu­ence on sub­sequent gen­er­a­tions of Amer­ic­an math­em­aticians, we next con­sider Dick­son as a lec­turer and ad­viser. What was he like in the classroom? What mo­tiv­ated his teach­ing style? How did he re­late to his doc­tor­al stu­dents? This in­form­a­tion serves to form a char­ac­ter­iz­a­tion of Dick­son as an edu­cat­or. With this in place, we take a closer look at the math­em­at­ic­al pur­suits of three of Dick­son’s wo­men stu­dents — Olive Hazlett, May­me Logs­don, and Mina Rees — to gain an un­der­stand­ing of both the spe­cif­ics of Dick­son’s al­geb­ra­ic her­it­age and his broad­er role in train­ing wo­men in re­search-level math­em­at­ics. Fi­nally, a care­ful con­sid­er­a­tion of Dick­son’s most suc­cess­ful pro­tege, A. Ad­ri­an Al­bert, brings con­firm­a­tion to both the con­sol­id­a­tion and the growth of the Chica­go re­search tra­di­tion.

Since this pa­per fo­cuses on the stu­dent — ad­viser re­la­tion­ship, we provide a so­ci­olo­gic­al frame­work for our as­sess­ment of Dick­son as an ad­viser and per­petu­at­or of a re­search tra­di­tion. In the last 30 or so years, so­ci­olo­gists and oth­ers have be­gun to con­sider this sig­ni­fic­ant aca­dem­ic re­la­tion­ship in an ef­fort to shed light on the dy­nam­ics of this key as­pect of both the trans­mis­sion of know­ledge and the de­vel­op­ment of the pro­fes­sion. As this im­port­ant av­en­ue for the ex­change of ideas at­trac­ted in­creas­ing at­ten­tion, terms such as “ment­or,” “role mod­el,” and “spon­sor” be­came a part of our lan­guage, and, con­sequently, our think­ing ([e36], pp. 693, 708–709). The pre­cise mean­ings of the terms may vary some­what from study to study but in gen­er­al a “role mod­el” de­scribes one who “pos­sesses skills and dis­plays tech­niques which the stu­dent lacks … and from whom, by ob­ser­va­tion and com­par­is­on with his own per­form­ance the stu­dent can learn” ([e36], p. 693). The term “ment­or” has come to refer to those “older people in an or­gan­iz­a­tion or pro­fes­sion who take young­er col­leagues un­der their wings and en­cour­age and sup­port their ca­reer pro­gress un­til they reach mid-life” ([e36], p. 708). or to in­dic­ate someone “high­er in the in­sti­tu­tion or or­gan­iz­a­tion who coaches, teaches, ad­vises, provides sup­port and guid­ance, and helps the mentee (what a dread­ful word!) achieve his or her goa[l]s” ([e46], p. 2). Thus, a role mod­el primar­ily teaches his or her trade by ex­ample, while a ment­or en­cour­ages the up-and-com­ing pro­fes­sion­al in a more per­son­al, in­ter­act­ive way. We will ad­opt this dis­tinc­tion here, while not­ing that it is, of course, en­tirely pos­sible for one per­son to serve sim­ul­tan­eously as a role mod­el and a ment­or.

Dickson’s mathematical roots

A seem­ingly un­likely man, the com­par­at­ive philo­lo­gist, Wil­li­am Rainey Harp­er, found him­self in a po­s­i­tion of con­sid­er­able in­flu­ence on Dick­son (and oth­ers, for that mat­ter) after he opened the doors of the Uni­versity of Chica­go for in­struc­tion in 1892. Harp­er’s own dis­cov­ery of aca­deme, however, had come some 15 years earli­er, giv­ing him an in­side view of the meta­morph­os­is of the Amer­ic­an uni­versity. Sub­stan­tial re­forms in the Amer­ic­an uni­versity sys­tem had oc­curred in the dec­ades fol­low­ing the close of the Civil War. The new de­vel­op­ments grew out of a dis­con­tent which had aris­en be­cause of an Amer­ic­an de­sire to match the es­tab­lished European uni­versity sys­tem, the avail­ab­il­ity of new wealth, and an in­creas­ing con­cern over the de­cline of Amer­ic­an col­legi­ate in­flu­ence ([e28], pp. 2–3). Pub­lic ser­vice and a com­mit­ment to ab­stract re­search (based on the Ger­man mod­el) emerged among the con­cep­tions of the new Amer­ic­an uni­versity ([e28], p. 12).

Aware of the chan­ging tide in Amer­ic­an high­er edu­ca­tion, Harp­er ex­pan­ded the ideas of his trust­ees and primary fin­an­cial back­er, John D. Rock­e­feller, by his in­sist­ence that the Uni­versity rise above the level of an un­der­gradu­ate col­lege by em­phas­iz­ing the train­ing of fu­ture re­search­ers. Des­pite Harp­er’s new design of a uni­versity, he did not in­tro­duce the first re­search-ori­ented in­sti­tu­tion on Amer­ic­an soil. Both the Johns Hop­kins Uni­versity, es­tab­lished in 1876, and Clark Uni­versity, foun­ded in 1889, had already at­temp­ted to em­phas­ize teach­ing as well as gradu­ate stud­ies and re­search ([e53], pp. 262–263).2 Harp­er drew from the stand­ards Pres­id­ents Daniel Co­it Gil­man of Johns Hop­kins and G. Stan­ley Hall of Clark had set in es­tab­lish­ing their in­sti­tu­tions in gen­er­al and in as­sem­bling their fac­ulties in par­tic­u­lar.

In Harp­er’s view, a pro­fess­or should “be a teach­er, but first and fore­most a schol­ar, in love with learn­ing, with a pas­sion for re­search, an in­vest­ig­at­or who could pro­duce, and, if what he pro­duced was worthy, would wish to pub­lish” ([e12], pp. 123–124). With his high stand­ards for fac­ulty mem­bers, Harp­er ten­ded to pur­sue well-es­tab­lished mem­bers of the aca­dem­ic world, in­clud­ing uni­versity pres­id­ents.3 He must have also re­cog­nized po­ten­tial, however, for Harp­er gave an un­proven as­so­ci­ate pro­fess­or from North­west­ern Uni­versity, Eliakim Hast­ings Moore, the tem­por­ary (and, ul­ti­mately, per­man­ent) reins of his ori­gin­al re­search-ori­ented math­em­at­ics de­part­ment. Moore’s ap­point­ment rep­res­en­ted a gamble on Harp­er’s part, but one which would have a “tre­mend­ous in­flu­ence” on the his­tory of Amer­ic­an math­em­at­ics ([e21], p. 3). Moore se­cured Os­kar Bolza, a pro­fess­or from the fledgling Clark Uni­versity, and Hein­rich Masch­ke, the Göttin­gen Ph.D. then work­ing as an elec­tri­cian in New Jer­sey, as his two first col­leagues. With Moore’s com­mit­ment to the highly suc­cess­ful Ger­man uni­versity tra­di­tion and the first-hand ex­per­i­ence of Bolza and Masch­ke with­in that sys­tem as stu­dents of Fe­lix Klein at Göttin­gen, the Chica­go math­em­at­ics de­part­ment opened its doors with its sights set on emu­lat­ing this tra­di­tion.4 Ger­many served as the mod­el of ex­cel­lence for vir­tu­ally all Amer­ic­ans in­ter­ested in math­em­at­ics dur­ing the last quarter of the 19th cen­tury. Moore him­self had ob­served the Ger­man uni­versity tra­di­tions dur­ing the year (1885–1886) he spent at the Uni­versit­ies of Göttin­gen and Ber­lin. Thus, Moore saw the Ger­man em­phas­is on pure re­search and be­came ac­quain­ted with the sem­in­ar, the ef­fect­ive Ger­man teach­ing tool. The sem­in­ar brought to­geth­er stu­dents and pro­fess­ors for the present­a­tion and dis­cus­sion of both ori­gin­al and re­cently pub­lished re­search. The real treas­ure of the sem­in­ar, however, lay in the “close hu­man con­tact … offered between ad­vanced stu­dents and a man of ma­jor repu­ta­tion in the field” ([e28], p. 156). For as­pir­ing math­em­aticians who traveled to Ger­many in the lat­ter part of the 20th cen­tury, the “men of repu­ta­tion” in­cluded Karl Wei­er­strass, Leo­pold Kro­neck­er, Fe­lix Klein, Sophus Lie, and later, Dav­id Hil­bert. With Amer­ic­an uni­versit­ies still largely ori­ented to­ward an un­der­gradu­ate pop­u­la­tion, the re­search-minded math­em­at­ics stu­dents ten­ded to look to Ger­many for their train­ing.

With their en­thu­si­asm for math­em­at­ics and their unique con­tri­bu­tions in terms of re­search and tal­ents, Moore, Bolza, and Masch­ke offered the steady stream of Amer­ic­an math­em­at­ics stu­dents oth­er­wise headed for Ger­many a reas­on to stay home. The trio es­tab­lished their suc­cess­ful de­part­ment through ef­fect­ive classroom teach­ing, the or­gan­iz­a­tion of the Math­em­at­ic­al Club for the re­view of books and mem­oirs and the present­a­tion of ori­gin­al re­search, and the ex­ample of top-qual­ity pub­lished work ([e53], pp. 371–372).5 This core of Moore, Bolza, and Masch­ke not only strove to de­vel­op a fine de­part­ment at Harp­er’s uni­versity, but they also played in­stru­ment­al roles in the es­tab­lish­ment of a more truly Amer­ic­an (as op­posed to New Eng­land) math­em­at­ic­al com­munity. Moore and the mid­west­ern math­em­aticians helped found a journ­al for ori­gin­al re­search (ul­ti­mately known as the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety) and pro­moted the de­vel­op­ment of vari­ous av­en­ues (Col­loquia, Sec­tions, and the Chica­go Con­gress, for ex­ample) to ex­change and stim­u­late math­em­at­ic­al thought.6 By tak­ing a broad per­spect­ive on Amer­ic­an math­em­at­ics, the Chica­go de­part­ment demon­strated to its stu­dents the im­port­ance of con­trib­ut­ing to the much lar­ger math­em­at­ic­al com­munity ([e53], pp. 401–419). Ow­ing to its suc­cess­ful im­ple­ment­a­tion and con­tinu­ation of a com­mit­ment to high re­search stand­ards, the Uni­versity of Chica­go held a unique po­s­i­tion among Amer­ic­an in­sti­tu­tions in the clos­ing dec­ade of the 19th cen­tury. Moore, Bolza, and Masch­ke sought to build and suc­ceeded in form­ing a math­em­at­ics de­part­ment which pro­moted ori­gin­al re­search, qual­ity pub­lic­a­tions, and a broad view of the Amer­ic­an math­em­at­ic­al com­munity. The strong in­sti­tu­tion­al and de­part­ment­al philo­sophy in­her­ent in these goals and mani­fest in their achieve­ment made the Uni­versity of Chica­go a vi­able op­tion for Le­onard Eu­gene Dick­son and oth­er as­pir­ing Amer­ic­an math­em­aticians, in­clud­ing Os­wald Veblen, Gil­bert Bliss, George D. Birk­hoff, and R. L. Moore, who might oth­er­wise have traveled to Ger­many for their train­ing.7 Born to Camp­bell and Lucy Tracy Dick­son in In­de­pend­ence, Iowa in 1874, Le­onard Dick­son made his home with his fam­ily in Texas, where his fath­er worked as a mer­chant and banker.8 Dick­son ob­tained his early edu­ca­tion in the pub­lic schools in his ho­met­own of Cleburne and at­ten­ded the Uni­versity of Texas for his un­der­gradu­ate and mas­ter’s edu­ca­tion. At the Uni­versity of Texas, Dick­son came un­der the in­flu­ence of “that ex­traordin­ary en­thu­si­ast for nonEuc­lidi­an geo­metry,” George Bruce Halsted.9 With his mas­ter’s de­gree in hand and two years of teach­ing ex­per­i­ence un­der his belt, Dick­son chose the strong Moore–Bolza–Masch­ke tri­um­vir­ate at Chica­go over the up-and-com­ing Har­vard with Wil­li­am Fogg Os­good and Maxime Bôcher as the place to pur­sue his doc­tor­ate.10 One can­not help but won­der how math­em­at­ics in Amer­ica might have been dif­fer­ent had Dick­son chosen Har­vard in­stead of Chica­go.

Dick­son headed for Chica­go as a nat­ive Tex­an pos­sess­ing what Gar­rett Birk­hoff de­scribed as “much of the dy­nam­ic en­ergy and rugged in­di­vidu­al­ism that we as­so­ci­ate with that state” [e34], p. 34. These char­ac­ter­ist­ics, along with his math­em­at­ic­al abil­ity, placed him among the tal­en­ted and per­sever­ing stu­dents left un­in­tim­id­ated by E. H. Moore’s teach­ing style.11 In fact, it ap­pears as though Moore’s un­usu­al ped­ago­gic­al ap­proaches, com­bined with his re­search in­terests in al­gebra and the found­a­tions of math­em­at­ics dur­ing the late 19th and early 20th cen­tur­ies, in­flu­enced Dick­son pro­foundly.

Dick­son ar­rived at Chica­go in 1894, just in time for Moore’s lec­tures on group the­ory. Dick­son wrote his dis­ser­ta­tion in group the­ory un­der the dir­ec­tion of Moore in 1896 [1]. The re­vised and ex­pan­ded ver­sion of this work ap­peared in 1901 in the form of his first book, writ­ten at the age of 27, Lin­ear Groups with an Ex­pos­i­tion of the Galois Field The­ory [14]. In the mean­time, Dick­son had traveled to Leipzig and Par­is where he stud­ied un­der Sophus Lie and Ca­m­ille Jordan, held a couple of brief ap­point­ments at the Uni­versit­ies of Cali­for­nia and Texas, and, in 1900, re­turned to the Uni­versity of Chica­go to be­gin his 40-year-long pro­fes­sion­al af­fil­i­ation with that school. As the 20th cen­tury opened, Dick­son again fol­lowed Moore’s lead — but now a Moore cap­tiv­ated by the ideas in­her­ent in Dav­id Hil­bert’s Found­a­tions of Geo­metry [e2] — and con­sidered found­a­tion­al ques­tions in math­em­at­ics. In­spired by his ad­viser, Dick­son pur­sued al­geb­ra­ic re­searches which would ul­ti­mately define his re­search for more than forty years. Yet this would mean al­gebra in its broad­est sense, for Dick­son would work in group the­ory, in­vari­ant the­ory, fi­nite field the­ory, and the the­ory of al­geb­ras. Dick­son re­flec­ted his ad­viser’s in­flu­ence in more ways than re­search in­terests, however. The de­part­ment’s sus­tained com­mit­ment to re­search, their high stand­ards for pub­lic­a­tion, and their vis­ion for the Amer­ic­an (as op­posed to New Eng­land) math­em­at­ic­al com­munity came to per­meate Dick­son’s math­em­at­ic­al per­sona in these form­at­ive years. In his 40-year ca­reer, Dick­son pub­lished 18 books and roughly 300 re­search art­icles, and, in so do­ing, made gradu­ate texts in math­em­at­ics read­ily avail­able to Amer­ic­an gradu­ate stu­dents. From his post as an ed­it­or of the Amer­ic­an Journ­al of Math­em­at­ics and the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety, he in­sisted upon qual­ity pub­lic­a­tions. He shared the broad vis­ion of the Chica­go math­em­at­ics de­part­ment, and, like Moore, served as pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety. In the pro­cess of for­ging ahead in math­em­at­ics and voicing frus­tra­tion over what he per­ceived as road­b­locks, he showed his stu­dents what he did — and did not — value as a pro­fes­sion­al math­em­atician. He thus per­petu­ated al­beit with some­what of a Dick­so­ni­an twist, the math­em­at­ic­al, aes­thet­ic, and pro­fes­sion­al val­ues modeled by the ori­gin­al tri­um­vir­ate at Chica­go.

Dickson’s mathematical legacy

Dick­son left a strong im­pres­sion on those who oc­cu­pied the chairs in his classroom or of­fice. Wit­ness, for ex­ample, the vivid de­scrip­tions stu­dents from over half a cen­tury ago give today of his math­em­at­ic­al per­son­al­ity. When they de­scribe Dick­son suc­cinctly, they tend to use terms like “gruff” ([e53], p. 377) or “hard-bit­ten” ([e49]) or “en­er­get­ic and force­ful” ([e40], p. 246). For this group, one need only men­tion the name of Dick­son and a flood of per­son­al re­col­lec­tions re­gard­ing his teach­ing and ad­vising style come to the fore.

A dis­cus­sion of those who em­ploy con­ven­tion­al ped­ago­gic­al ap­proaches would cer­tainly not in­clude Dick­son. He de­livered terse and un­pol­ished lec­tures and spoke sternly to his stu­dents. He fre­quently as­signed read­ings from a text­book (of­ten one of his own), and he either called on stu­dents to present and ana­lyze the ma­ter­i­al, or he lec­tured the en­tire hour. This meth­od mo­tiv­ated stu­dents to do their best in mak­ing classroom present­a­tions [e48]. Giv­en Dick­son’s in­tol­er­ance for stu­dent weak­nesses in math­em­at­ics, however, his com­ments could be harsh, even though not in­ten­ded to be per­son­al. He did not aim to make his stu­dents feel good about them­selves [e50]. In fact, in the words of one of his former stu­dents, “[h]e was blunt and straight­for­ward about his ex­pect­a­tions from his stu­dents. He him­self worked very hard and he ex­pec­ted his stu­dents to work hard too” [e50]. These gen­er­al ped­ago­gic­al com­ments about Dick­son hardly hint at a ment­or in the sense of someone high­er in the in­sti­tu­tion who took his stu­dents “un­der his wings” and “coached” or “provided sup­port and guid­ance” as they pur­sued their goals. In spite of his “gruff” ap­proach, however, Dick­son did suc­cess­fully serve as a role mod­el as he im­par­ted al­gebra and num­ber the­ory to the next gen­er­a­tion of Amer­ic­an math­em­aticians. Wil­li­am Duren, a Chica­go Ph.D. stu­dent in the late twen­ties and early thirties, un­der­scored the nature of Dick­son’s brand of mod­el­ing. “In the con­ven­tion­al sense,” Duren re­flec­ted, “Dick­son was not much of a teach­er. I think his stu­dents learned from him by emu­lat­ing him as a re­search math­em­atician more than be­ing taught by him. Moreover, he took them to the fron­ti­ers of re­search, for the sub­ject mat­ter of his courses was usu­ally new math­em­at­ics in the mak­ing. As Ant­oinette Hu­s­ton [a gradu­ate stu­dent at Chica­go] said, ‘Dick­son made you want to be with him in­tel­lec­tu­ally. When you are young, reach­ing for the stars, that is what it is all about’ ” ([e52], p. 178).

Dick­son’s teach­ing, it seems, re­flec­ted his lifelong goal to be­come the most dis­tin­guished math­em­atician pos­sible. He spent his math­em­at­ic­al life at the cut­ting edge of the field, and he wanted his stu­dents to do the same. Since stu­dents who could not meet his stand­ards also could not serve his pur­poses best, Dick­son had a sud­den death tri­al for his pro­spect­ive doc­tor­al stu­dents: he as­signed a pre­lim­in­ary prob­lem which was short­er than a dis­ser­ta­tion prob­lem, and if the stu­dent could solve it in three months, Dick­son would agree to over­see the gradu­ate stu­dent’s work. If not, the stu­dent had to look else­where for an ad­viser ([e45], p. 377). Dick­son quite clearly de­signed his three-month test prob­lem as a means of eval­u­at­ing wheth­er or not a stu­dent could make the trip to the “fron­ti­ers of re­search,” and al­though some stu­dents may not have cared for this ap­proach, 67 Chica­go Ph.D. stu­dents ac­cep­ted it enough to sign on for his guid­ance. As a com­par­is­on, Gil­bert Bliss, the oth­er prin­cip­al ad­viser of math­em­at­ics Ph.D.’s at Chica­go in the early dec­ades of this cen­tury, over­saw 46 doc­tor­al stu­dents.

A look at the faces be­hind the dis­ser­ta­tions dir­ec­ted by Dick­son un­cov­ers the rich and col­or­ful leg­acy left hid­den by a merely nu­mer­ic­al count of his stu­dents. In par­tic­u­lar, wo­men ac­coun­ted for 18 (roughly 27%) of these stu­dents. Set with­in the broad­er con­text of wo­men in math­em­at­ics, these eight­een Ph.D.’s com­posed slightly more than 8% of all wo­men who earned an Amer­ic­an Ph.D. in math­em­at­ics between 1900 and 1939 and 40% of those awar­ded to wo­men at the Uni­versity of Chica­go.12 Olive Hazlett, a 1915 Dick­son doc­tor­ate, fol­lowed her ad­viser’s lead and stud­ied nil­po­tent al­geb­ras, di­vi­sion al­geb­ras, and mod­u­lar in­vari­ants [e11], [e13], [e6], [e8], [e7], [e10], [e9], be­fore she ap­par­ently fol­lowed Dick­son down the road to the arith­met­ic of al­geb­ras. At the 1924 In­ter­na­tion­al Con­gress of Math­em­aticians in Toronto, she ex­ten­ded his arith­met­ic of ra­tion­al al­geb­ras to al­geb­ras over an ar­bit­rary field [e13]. Hazlett could not claim the ex­clus­ive rights for these ideas, however, since Dick­son presen­ted the same gen­er­al­iz­a­tion at the same meet­ing [10]. (This, per­haps, rep­res­en­ted a bit­ter­sweet mo­ment for Dick­son (and Hazlett, for that mat­ter), bit­ter in that he was not the only one present­ing a more gen­er­al the­ory but sweet in that he had trained his stu­dent well.) Like Dick­son, Hazlett cer­tainly placed a high value on re­search math­em­at­ics. She wrote more pa­pers than any oth­er pre-1940 Amer­ic­an wo­man math­em­atician ([e44], p. 138) and, re­flect­ive of her ad­viser’s com­mit­ment to the broad­er math­em­at­ic­al com­munity, she served as a co­oper­at­ing ed­it­or of the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety for 12 years, held a two-year term on the AMS Coun­cil, and she, Char­lotte Scott, and Anna John­son Pell Wheel­er com­posed the en­tire group of wo­men math­em­aticians starred in Amer­ic­an Men of Sci­ence between 1903 and 1943 ([e38], p. 293). As for her re­la­tion­ship with Dick­son, it can be char­ac­ter­ized as one of mu­tu­al re­spect. In par­tic­u­lar, al­though she con­tin­ued to draw from his work, Hazlett ap­par­ently neither ten­ded to con­sult Dick­son dir­ectly about her em­ploy­ment di­lem­mas nor to keep him in­formed of her re­search.13 In short, for Hazlett, Dick­son served as a role mod­el rather than a ment­or.

May­me Logs­don, a 1921 Dick­son Ph.D., who sub­sequently be­came in­ter­ested in al­geb­ra­ic geo­metry, served on the Uni­versity of Chica­go fac­ulty for 25 years. In terms of the broad­er em­ploy­ment pic­ture for wo­men in math­em­at­ics, Logs­don rep­res­en­ted what one might call a typ­ic­al an­om­aly. Her situ­ation was an­om­al­ous in the sense that she en­joyed one of the few po­s­i­tions offered to a wo­man by a ma­jor re­search in­sti­tu­tion.14 It was typ­ic­al in that she gen­er­ally taught un­der­gradu­ate courses and nev­er re­ceived pro­mo­tion to full pro­fess­or dur­ing her many years of ser­vice to Chica­go. Nev­er­the­less, Logs­don con­tin­ued to pur­sue re­search, which in­cluded spend­ing an en­tire aca­dem­ic year (1925–1926) in Italy when it was the most act­ive cen­ter of re­search for al­geb­ra­ic geo­metry. She main­tained a high enough level of re­search to over­see the work of four Ph.D. stu­dents.15 Thus, Logs­don, like Hazlett, pur­sued an aca­dem­ic ca­reer which em­phas­ized many of the qual­it­ies modeled by Dick­son.

Mina Rees, a 1931 Dick­son doc­tor­ate, re­flec­ted her ad­viser’s in­flu­ence in a dif­fer­ent way. Like Dick­son, she had a vis­ion for the Amer­ic­an com­munity of re­search math­em­aticians. Al­though Dick­son cer­tainly held im­port­ant ad­min­is­trat­ive po­s­i­tions — such as the pres­id­ency of the AMS and the ed­it­or­ships of the Trans­ac­tions and the Amer­ic­an Journ­al — he made his most sig­ni­fic­ant con­tri­bu­tions to the de­vel­op­ment of math­em­at­ics in this coun­try by way of his re­search and pub­lic­a­tions. Rees, on the oth­er hand, largely left re­search math­em­at­ics be­hind her after the com­ple­tion of her doc­tor­ate [e16] and wiel­ded her greatest in­flu­ence through her policy-set­ting lead­er­ship roles. She ad­vanced the cause of Amer­ic­an math­em­at­ics primar­ily by se­cur­ing fed­er­al funds for the field, re­cog­niz­ing (in the 1950s) the im­port­ance of com­puters in sci­entif­ic de­vel­op­ment, and ad­dress­ing the chan­ging needs of math­em­at­ics edu­ca­tion. For Rees, then (and per­haps oth­ers?), Dick­son seems to have served as a role mod­el of lead­er­ship.

In ad­di­tion to Logs­don, Hazlett, and Rees, Dick­son ad­vised 15 oth­er wo­men stu­dents.16 In so do­ing, he dir­ec­ted more than one-fifth of all Amer­ic­an al­gebra Ph.D.’s gran­ted to wo­men be­fore 1940 and all but one of the dis­ser­ta­tions in num­ber the­ory ([e39], p. 20). Ac­cord­ing to Judy Green and Jeanne LaDuke, Dick­son served as something of a “cluster point” for as­pir­ing wo­men math­em­aticians ([e39], p. 20). Alice Schafer has fur­ther sug­ges­ted that a snow­ball ef­fect may have led to Dick­son’s large num­ber of wo­men gradu­ate stu­dents. In oth­er words, once a few wo­men com­pleted their de­grees un­der his guid­ance, fa­vor­able word spread about his abil­it­ies as an ad­viser [e51]. The Uni­versity of Chica­go may have also con­trib­uted to Dick­son’s de­scrip­tion as a “cluster point” for wo­men. Dick­son’s stu­dent, Gwen­eth Humphreys, noted that the school had a fa­vor­able repu­ta­tion among wo­men seek­ing high­er de­grees in math­em­at­ics. Fol­low­ing the start of their stud­ies there, they, like so many of the men, found Dick­son among the most — if not the most — com­mit­ted re­search math­em­atician on the fac­ulty and se­lec­ted their ad­viser based on this cri­terion.17 Thus, per­haps as a res­ult of his early repu­ta­tion as an ad­viser for wo­men, his in­sti­tu­tion, and his strong re­search pro­gram, Dick­son evolved in­to a not­ably suc­cess­ful ad­viser for wo­men pur­su­ing math­em­at­ics doc­tor­ates in the United States. As the re­marks about his ped­ago­gic­al style and the ca­reers of the wo­men re­coun­ted above re­veal, however, his suc­cess as an ad­viser lay primar­ily in his abil­ity to im­part his trade by ex­ample rather than to en­cour­age his stu­dents in a more per­son­al, in­ter­act­ive way.

Even still, cer­tain as­pects of Dick­son’s per­sona as an ad­viser to wo­men re­main un­clear. Were there oth­er reas­ons for which Dick­son be­came a “cluster point” for wo­men gradu­ate stu­dents at Chica­go? Did he have per­son­al mo­tiv­a­tions for want­ing to dir­ect wo­men? Even with tra­di­tion­al ste­reo­types taken in­to con­sid­er­a­tion, a “hard-bit­ten” math­em­at­ic­al per­son­al­ity does not seem the most likely mag­net for as­pir­ing wo­men (or men?) math­em­aticians. Did the wo­men doc­tor­al stu­dents in the early dec­ades of this cen­tury pos­sess a cer­tain amount of forti­tude which im­mun­ized them against dif­fi­cult per­son­al­ity traits and kept them fo­cused on their goal?18 Even with these ques­tions out­stand­ing, this study of Dick­son as an ad­viser clearly un­cov­ers his note­worthy con­tri­bu­tions to the edu­ca­tion of wo­men math­em­aticians in the open­ing dec­ades of this cen­tury as well as to the 20th-cen­tury her­it­age of wo­men in re­search-level math­em­at­ics in the United States. Moreover, the ex­per­i­ences of Hazlett, Logs­don, and Rees demon­strate that Dick­son served as a role mod­el in at least two ways — as a re­search­er and as a com­munity-minded act­iv­ist. Thus, the strong re­search eth­ic and the vis­ion for the Amer­ic­an math­em­at­ic­al com­munity which Moore had modeled for Dick­son, be­came more firmly en­trenched in the Chica­go al­geb­ra­ic tra­di­tion. The dir­ect con­tinu­ation of this her­it­age at Chica­go, however, lay in the hands of an­oth­er of Dick­son’s doc­tor­al stu­dents, A. A. Al­bert.

Al­bert, or A-cubed as he was of­ten called, ar­rived at the Uni­versity of Chica­go in 1922 when the the­ory of al­geb­ras was among Dick­son’s main re­search in­terests. Dick­son’s “con­sid­er­able” in­flu­ence ([e40], p. 246) on Al­bert mani­fes­ted it­self in his 1927 mas­ter’s thes­is where he de­term­ined all 2-, 3-, and 4-di­men­sion­al as­so­ci­at­ive al­geb­ras over a non­mod­u­lar field \( F \) and in his 1928 dis­ser­ta­tion en­titled “Al­geb­ras and Their Rad­ic­als, and Di­vi­sion Al­geb­ras.” In the lat­ter, Al­bert proved that every cent­ral di­vi­sion al­gebra of di­men­sion 16 is not ne­ces­sar­ily cyc­lic but is al­ways a crossed product. Al­bert pol­ished this work and presen­ted it as his first ma­jor pub­lic­a­tion [e14]. Irving Ka­plansky com­men­ted on the math­em­at­ic­al per­son­al­ity Al­bert re­vealed in his early work when he wrote: “Here was a tough prob­lem that had de­feated his pre­de­cessors; he at­tacked it with tenacity till it yiel­ded” ([e40], p. 247).

Al­bert’s thes­is re­search on cent­ral di­vi­sion al­geb­ras placed him at the cen­ter of activ­ity in the field of lin­ear as­so­ci­at­ive al­geb­ras. In par­tic­u­lar, he, along with the Ger­man math­em­aticians Richard Brauer, Helmut Hasse, and Emmy No­eth­er, strove to de­term­ine all cent­ral di­vi­sion al­geb­ras. In 1931, the Ger­man trio reached the cent­ral di­vi­sion al­gebra fin­ish line only seconds be­fore a breath­less Al­bert. They es­tab­lished the prin­cip­al the­or­em that every cent­ral di­vi­sion al­gebra over an al­geb­ra­ic num­ber field of fi­nite de­gree is cyc­lic [e17]. One year later, Al­bert and Hasse pub­lished a joint work [e15] which gave the his­tory of the the­or­em and de­scribed Al­bert’s “near miss” ([e30], p. 663). Al­though Al­bert would go on to make sig­ni­fic­ant con­tri­bu­tions to the the­ory of Riemann matrices [e18], [e20], [e19] and to in­tro­duce single­han­dedly the Amer­ic­an school of nonas­so­ci­at­ive al­geb­ras [e23], he main­tained an in­terest in as­so­ci­at­ive di­vi­sion al­geb­ras throughout his more than 40-year-long ca­reer.

The scope of Al­bert’s tal­ents ex­ten­ded far bey­ond the pro­duc­tion and pub­lic­a­tion of math­em­at­ic­al res­ults. He, like Dick­son and Moore, made sig­ni­fic­ant con­tri­bu­tions to both the Uni­versity of Chica­go and the Amer­ic­an Math­em­at­ic­al So­ci­ety. Re­l­at­ive to the former, the mem­bers of the math­em­at­ics de­part­ment at Chica­go — Dick­son, Bliss, and E. H. Moore, among them — re­cog­nized Al­bert’s abil­it­ies and se­cured him as a per­man­ent fac­ulty mem­ber soon after he com­pleted his Ph.D. It was, in fact, more than Al­bert’s abil­ity which landed him the as­sist­ant pro­fess­or­ship — and ob­vi­ous po­s­i­tion as heir ap­par­ent to Dick­son — in 1931, however. In­deed, dur­ing its first four dec­ades, the pro­ced­ure for hir­ing in the Chica­go math­em­at­ics de­part­ment seemed to fol­low what Saun­ders Mac Lane has called an “in­her­it­ance prin­ciple” ([e43], p. 141). This ap­point­ment pro­ced­ure worked ex­cep­tion­ally well as Moore, Dick­son, and Al­bert es­tab­lished a strong al­geb­ra­ic tra­di­tion at Chica­go.

Dur­ing Al­bert’s ten­ure as a fac­ulty mem­ber at his alma ma­ter, he par­ti­cip­ated in a vari­ety of com­mit­tees, or­gan­ized con­fer­ences, chaired the math­em­at­ics de­part­ment, and served as the dean of the Di­vi­sion of Phys­ic­al Sci­ences. While chair, he “skill­fully” found sup­port to main­tain a steady flow of vis­it­ors and re­search in­struct­ors ([e40], pp. 251–252) Al­bert used his in­flu­ence to per­suade the Uni­versity to donate an apart­ment build­ing, af­fec­tion­ately known as “the com­pound,” to house the vis­it­ors. Ka­plansky claims that “the com­pound” be­came the “birth­place of many a fine the­or­em” ([e40], p. 252). In par­tic­u­lar, two vis­it­ors call­ing it home in 1960–1961, Wal­ter Feit and John Thompson, de­term­ined that all groups of odd or­der are solv­able ([e40], p. 252). When con­sid­er­ing the sum total of Al­bert’s math­em­at­ic­al ca­reer, it should come as no sur­prise that he strove to at­tract vis­it­ors to Chica­go. He real­ized that a de­part­ment which re­lied solely on its per­man­ent fac­ulty had the po­ten­tial to be­come stale and nar­row in its fo­cus. An in­filt­ra­tion of new ideas fre­quently en­cour­aged a fresh per­spect­ive on math­em­at­ics. Al­bert re­cog­nized that math­em­at­ic­al pro­gress of­ten de­pended on just such an ex­tern­al spark.

Al­bert’s ca­reer also re­flec­ted a strong com­mit­ment to the math­em­at­ic­al com­munity at large. He served the AMS in a vari­ety of ca­pa­cit­ies — as a com­mit­tee mem­ber, as an ed­it­or of the Bul­let­in and Trans­ac­tions, and, like Dick­son and Moore, as Pres­id­ent in 1965–1966. The con­cerns of Amer­ic­an math­em­aticians in the middle two quar­ters of the twen­ti­eth cen­tury were, however, some­what dif­fer­ent from those in the early years when Moore and Dick­son made their con­tri­bu­tions, and Al­bert’s ser­vice quite nat­ur­ally ad­dressed the chan­ging needs of Amer­ic­an math­em­aticians. In par­tic­u­lar, Al­bert helped es­tab­lish gov­ern­ment re­search grants for math­em­at­ics com­par­able to those ex­ist­ing in oth­er areas of sci­ence ([e32], p. 1077). He helped set the Na­tion­al Sci­ence Found­a­tion (NSF) budget for math­em­at­ics and aided in the cre­ation of the NSF sum­mer re­search in­sti­tutes ([e32], p. 1077). He ap­par­ently found sat­is­fac­tion in this na­tion­ally ori­ented work for “[h]e was al­ways pleased to use his in­flu­ence in Wash­ing­ton to im­prove the status of math­em­aticians in gen­er­al, and he was will­ing to do the same for in­di­vidu­al math­em­aticians whom he con­sidered worthy” ([e30], p. 665). This lat­ter cat­egory surely in­cluded his stu­dents.

Bey­ond his ser­vice to Chica­go, the AMS, and the math­em­at­ic­al com­munity at large, Al­bert ex­er­ted con­sid­er­able in­flu­ence in math­em­at­ics through his stu­dents ([e32], p. 1078). As his col­league, I. N. Her­stein, ob­served, “Ad­ri­an was ex­tremely good at work­ing with stu­dents. This is at­tested by the 30 math­em­aticians who took their Ph.D.’s with him. In their num­ber are many who are well known math­em­aticians today. His in­terest in his stu­dents — while they were stu­dents and forever af­ter­wards — was known and ap­pre­ci­ated by them” ([e31], p. 186). Daniel Zel­in­sky, in par­tic­u­lar, de­scribed Al­bert as an ad­viser who treated his Ph.D. stu­dents “al­most as mem­bers of his fam­ily” ([e30], p. 663). Al­bert’s stu­dents and col­leagues re­garded him warmly, a lux­ury his own ad­viser had not of­ten — if ever — known.

From a broad­er per­spect­ive, Al­bert in many ways ful­filled the role of an “heir” in that his math­em­at­ic­al ca­reer looked very much like Dick­son’s. Both re­ceived their Ph.D.’s from Chica­go, had in­flu­en­tial postdoc­tor­al years, held a few short ap­point­ments at schools oth­er than Chica­go, and ul­ti­mately spent the bulk of their ca­reers lead­ing the al­gebra pro­gram at their alma ma­ter. They both led seem­ingly tire­less math­em­at­ic­al lives, pro­du­cing — and ex­ert­ing a world­wide in­flu­ence through — ex­cep­tion­ally large num­bers of pub­lic­a­tions and gradu­ate stu­dents. (In­ter­est­ingly, a count of Al­bert’s pub­lic­a­tions and gradu­ate stu­dents rounds out at roughly half of Dick­son’s in all cat­egor­ies. Al­bert pub­lished eight books, and Dick­son 18. Al­bert wrote just over 140 art­icles and Dick­son just un­der 300. Al­bert ad­vised 30 Ph.D. stu­dents, and Dick­son 67.) They both ed­ited the Trans­ac­tions, served as AMS pres­id­ent, de­livered plen­ary lec­tures at In­ter­na­tion­al Math­em­at­ic­al Con­gresses, and re­ceived the AMS Cole Prize. Their col­leagues (in­de­pend­ently) de­scribed each of them as power­ful math­em­aticians. (On a less form­al note, Dick­son and Al­bert both took bil­liards ser­i­ously.) These strik­ing sim­il­ar­it­ies seem to sup­port the more con­tem­por­ary (1981) re­search of Black­burn, Chap­man, and Camer­on re­gard­ing the idea of “clon­ing” in aca­deme. In their study of ment­or­ship,19 they found that “[m]entors over­whelm­ingly nom­in­ated as their most suc­cess­ful pro­teges those whose ca­reers were es­sen­tially identic­al to their own — i.e., their ‘clones’ ” ([e35], p. 315). Al­though we have no writ­ten evid­ence that Dick­son ac­know­ledged Al­bert as his “best” stu­dent, Al­bert’s ca­reer, the “in­her­it­ance prin­ciple” for hir­ing at Chica­go, and the de­scrip­tions by Birk­hoff and Mac Lane sub­stan­ti­ate this idea. Best stu­dent or not, Al­bert echoed the pro­fes­sion­al heart­beat of his ad­viser — and role mod­el — throughout his long and dis­tin­guished ca­reer. Yet des­pite their many sim­il­ar­it­ies in the math­em­at­ic­al world, as per­son­al­it­ies Al­bert and Dick­son hardly re­sembled one an­oth­er at all.

Dick­son, ap­par­ently, had few close re­la­tion­ships with­in math­em­at­ics, where­as Al­bert en­joyed many close friends with­in the math­em­at­ic­al com­munity. Wit­ness, for ex­ample, the four warm bio­graph­ies writ­ten upon his death by Her­stein, Nath­an Jac­ob­son, Ka­plansky, and Zel­in­sky. After all, in ad­di­tion to his re­search work, Al­bert had ded­ic­ated him­self to such “homey” causes as ob­tain­ing a res­id­ence for the vis­it­ing math­em­aticians at Chica­go, and his stu­dents were well aware of his in­terest in them while un­der his guid­ance and later while they pur­sued their ca­reers. “Every­one who knew him,” Zel­in­sky wrote of Al­bert, “will re­mem­ber his vig­or­ous but round, me­di­um build, curly hair, and of­ten boy­ish de­mean­or; but es­pe­cially one must re­mem­ber his great, pleased grin that he flashed to wel­come news of new suc­cesses for any of his ex­ten­ded fam­ily any­where in the world of math­em­at­ics” ([e30], p. 665). Dick­son’s stu­dents did not re­call him in such en­dear­ing terms, but they did re­cog­nize the im­port­ance of what he im­par­ted to them. Un­like his ad­viser, Al­bert was a role mod­el and a ment­or.


When Pres­id­ent Harp­er opened the doors of the Uni­versity of Chica­go in 1892, he had his sights set on build­ing an in­sti­tu­tion which pro­moted schol­arly re­search and teach­ing. His gamble on the un­proven E. H. Moore yiel­ded a high re­turn as Moore and his Ger­man col­leagues, Bolza and Masch­ke, suc­ceeded in their quest to emu­late the Ger­man re­search tra­di­tion in the math­em­at­ics de­part­ment. The three quickly shaped their de­part­ment in­to an ef­fect­ive train­ing ground for fu­ture re­search math­em­aticians. They im­par­ted a strong re­search eth­ic — along with a com­mit­ment to both qual­ity pub­lic­a­tions and the broad­er math­em­at­ic­al com­munity — to the first gen­er­a­tion of Chica­go-trained math­em­aticians, not the least of whom was Le­onard Eu­gene Dick­son.

As the qual­it­ies Dick­son saw in Moore, Bolza, and Masch­ke began to shape his own math­em­at­ic­al per­sona, Dick­son spe­cific­ally pur­sued al­geb­ra­ic and num­ber-the­or­et­ic in­terests and gen­er­ally em­phas­ized far-reach­ing the­or­ies built on firm math­em­at­ic­al found­a­tions. Moreover, like the ori­gin­al tri­um­vir­ate at Chica­go, Dick­son im­pressed these and oth­er math­em­at­ic­al and ex­tramathem­at­ic­al qual­it­ies upon the second gen­er­a­tion of Chica­go-trained Ph.D.’s. Thus, in the first four dec­ades of the 20th cen­tury, Dick­son, as an act­ive role mod­el, helped define, fur­ther de­vel­op, and con­tin­ue the strong al­geb­ra­ic re­search tra­di­tion be­gun by Moore in the last dec­ade of the 19th cen­tury.

In the 40 years he served on the Chica­go fac­ulty, Dick­son ad­vised 67 doc­tor­al stu­dents in al­gebra and num­ber the­ory. His “gruff” per­son­al­ity and his al­most im­per­son­al in­ter­ac­tions with his stu­dents, to­geth­er with his high stand­ards for re­search and pub­lic­a­tion, clearly sug­gest this late 20th-cen­tury char­ac­ter­iz­a­tion of Dick­son as a role mod­el as op­posed to a ment­or. To be sure, Dick­son had qual­it­ies worth emu­lat­ing, but his math­em­at­ic­al per­sona com­bined with the gen­er­al aca­dem­ic cli­mate of his day fostered a more dis­tant ad­viser-stu­dent re­la­tion­ship than one might ex­pect today.

Dis­tant or oth­er­wise, as the ex­per­i­ences of the stu­dents re­coun­ted in this pa­per re­veal, Dick­son showed his stu­dents (and col­leagues, for that mat­ter) what the life of a pro­fes­sion­al math­em­atician could and should be. Wheth­er in their com­mit­ment to the pur­suit of math­em­at­ic­al re­search (Logs­don, Hazlett, and, es­pe­cially, Al­bert) or to the goals of the broad­er math­em­at­ic­al com­munity (Rees and Al­bert), these stu­dents re­flec­ted their ad­viser’s at­trib­utes. Fur­ther­more, their ca­reers fit se­curely with­in the al­geb­ra­ic re­search tra­di­tion which grew and prospered at Chica­go un­der Dick­son. All of his stu­dents took the Dick­so­ni­an gos­pel with them as they spread across the coun­try in the middle third of the 20th cen­tury to no few­er than 45 aca­dem­ic in­sti­tu­tions in at least 22 states and three for­eign coun­tries. They, like the par­tic­u­lar stu­dents dis­cussed in de­tail above, did work in the same spir­it as Dick­son — wheth­er in re­search, pub­lic­a­tion, or lead­er­ship po­s­i­tions — and this, after all, is how re­search tra­di­tions form and con­tin­ue.

What Moore had be­gun in the last dec­ade of the 19th cen­tury with his keen eye for re­search trends and his vis­ion for the broad­er math­em­at­ic­al com­munity, Dick­son shaped and furthered through the early dec­ades of the 20th cen­tury with his tire­less com­mit­ment to re­search, gradu­ate stu­dents, and qual­ity pub­lic­a­tions. Dick­son’s stu­dents — es­pe­cially Al­bert — en­sured the con­tinu­ation of this strong al­geb­ra­ic tra­di­tion at Chica­go and else­where as their em­bod­i­ment of this her­it­age mani­fes­ted it­self in their var­ied ca­reers. Thus, an in­vest­ig­a­tion of the whole of Dick­son’s math­em­at­ic­al ca­reer through the lens of role mod­el­ing — from those who modeled the life of a pro­fes­sion­al math­em­atician to him, to those for whom he modeled such a life — gives strong evid­ence that a math­em­at­ic­al com­munity can­not achieve its po­ten­tial for growth solely from re­search and pub­lic­a­tions. These ideas, tools, and val­ues must be fruit­fully cul­tiv­ated in the next gen­er­a­tion of math­em­aticians for the math­em­at­ics con­stitu­ency not only to con­tin­ue but also to flour­ish. In short, Dick­son’s re­search and pub­lic­a­tions of all types, coupled with an en­trench­ment of his math­em­at­ic­al and ex­tramathem­at­ic­al val­ues by his stu­dents as they dis­persed across the coun­try, helped to con­sol­id­ate and strengthen Amer­ic­an math­em­at­ics in the open­ing dec­ades of the 20th cen­tury.


[1]L. E. Dick­son: “The ana­lyt­ic rep­res­ent­a­tion of sub­sti­tu­tions on a power of a prime num­ber of let­ters with a dis­cus­sion of the lin­ear group, I: Ana­lyt­ic rep­res­ent­a­tion of sub­sti­tu­tions,” Ann. Math. 11 : 1–​6 (1896–1897), pp. 161–​183. Part II pub­lished in Ann. Math. 11:1–6 (1896–1897). See also Dick­son’s PhD thes­is. MR 1502221 article

[2]L. E. Dick­son: “Defin­i­tions of a lin­ear as­so­ci­at­ive al­gebra by in­de­pend­ent pos­tu­lates,” Trans. Am. Math. Soc. 4 : 1 (1903), pp. 21–​26. MR 1500620 JFM 34.​0090.​02 article

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

[4]L. E. Dick­son: “On hy­per­com­plex num­ber sys­tems,” Trans. Am. Math. Soc. 6 : 3 (1905), pp. 344–​348. MR 1500716 JFM 36.​0139.​02 article

[5]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

[6]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

[7]L. E. Dick­son: “A new simple the­ory of hy­per­com­plex in­tegers,” J. Math. Pure Ap­pl. 2 : 9 (1923), pp. 281–​326. An ab­stract was pub­lished in Bull. Am. Math. Soc. 29:3 (1923). JFM 49.​0089.​01 article

[8]L. E. Dick­son: “Al­geb­ras and their arith­met­ics,” Bull. Am. Math. Soc. 30 : 5–​6 (1924), pp. 247–​257. MR 1560885 JFM 50.​0631.​02 article

[9]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

[10]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

[11]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

[12]G. A. Bliss: “Eliakim Hast­ings Moore,” Bull. Amer. Math. Soc. 39 : 11 (1933), pp. 831–​838. MR 1562740 JFM 59.​0038.​02

[13] G. A. Bliss and L. E. Dick­son: “Eliakim Hast­ings Moore: 1862–1932,” Bio­graph­ic­al Mem­oirs of the Na­tion­al Academy of Sci­ences 17 (1937), pp. 83–​102. article

[14]L. E. Dick­son: Lin­ear groups with an ex­pos­i­tion of the Galois field the­ory, re­print edition. Dover Pub­lic­a­tions (New York), 1958. With an in­tro­duc­tion by Wil­helm Mag­nus. Re­pub­lic­a­tion of 1901 ori­gin­al. MR 0104735 Zbl 0082.​24901 book

[15]L. E. Dick­son: Lin­ear al­geb­ras, re­print edition. Hafn­er (New York), 1960. Re­pub­lic­a­tion of 1914 ori­gin­al. MR 0118745 book