THE ORIGINS OF THE GENUS CONCEPT IN QUADRATIC FORMS

Transcription

1 The Matheatcs Enthusast Volue 6 Nuber Nubers & Artcle THE ORIGINS OF THE GENUS CONCEPT IN QUADRATIC FORMS Mark Bentea Azar Khosravan Let us know how access to ths docuent benets you. Follow ths and addtonal works at: htts://scholarworks.ut.edu/te Part o the Matheatcs Coons Recoended Ctaton Bentea, Mark and Khosravan, Azar (009) "THE ORIGINS OF THE GENUS CONCEPT IN QUADRATIC FORMS," The Matheatcs Enthusast: Vol. 6 : No., Artcle 3. Avalable at: htts://scholarworks.ut.edu/te/vol6/ss/3 Ths Artcle s brought to you or ree and oen access by ScholarWorks at Unversty o Montana. It has been acceted or ncluson n The Matheatcs Enthusast by an authorzed edtor o ScholarWorks at Unversty o Montana. For ore noraton, lease contact scholarworks@so.ut.edu.

2 TMME, vol6, nos.&,.37 THE ORIGINS OF THE GENUS CONCEPT IN QUADRATIC FORMS Mark Bentea & Azar Khosravan College o Lake County, Illnos Coluba College Chcago ABSTRACT: We resent an eleentary exoston o genus theory or ntegral bnary quadratc ors, laced n a hstorcal context. KEY WORDS: Quadratc Fors, Genus, Characters AMS Subject Classcaton: 0A50, 0A55 and E6. INTRODUCTION: Gauss once aously rearked that atheatcs s the queen o the scences and the theory o nubers s the queen o atheatcs. Publshed n 80, Gauss Dsqustones Arthetcae stands as one o the crownng acheveents o nuber theory. The theory o bnary quadratc ors occues a large swath o the Dsqustones; one o the unyng deas n Gauss develoent o quadratc ors s the concet o genus. The generatons ollowng Gauss generalzed the concets o genus and class grou ar beyond what Gauss had done, and students aroachng the subject today can easly lose sght o the basc dea. Our goal s to gve a heurstc descrton o the concet o genus accessble to those wth lted background n nuber theory and lace t n a hstorcal context. We do not retend to gve the ost general treatent o the toc, but rather to show how the dea orgnally develoed and how Gauss orgnal denton les the ore coon denton ound n today s texts., Mark Bentea Deartent o Matheatcs College o Lake County 935 W. Washngton Ave. 600 S. Mchgan Ave. Grayslake, IL Chcago, IL Azar Khosravan Scence and Math Deartent Coluba College Chcago (87) (3) arkbentea@clcllnos.edu akhosravan@colu.edu The Montana Matheatcs Enthusast, ISSN 55-30, Vol. 6, nos.&, Montana Councl o Teachers o Matheatcs & Inoraton Age Publshng

3 Bentea & Khosravan BASIC DEFINITIONS: An ntegral bnary quadratc or s a olynoal o the tye ( x, cy ax bxy, where a, b, and c are ntegers. A or s rtve the ntegers a, b, and c are relatvely re. Note that any or s an nteger ultle o a rtve or. Throughout, we wll assue that all ors are rtve. We say that a or reresents an nteger n ( x, n has an nteger soluton; the reresentaton s roer the ntegers x, y are relatvely re. A or s ostve dente t reresents only ostve ntegers; we wll restrct our dscusson to ostve dente ors. that The dscrnant o ax bxy cy s dened as b ac (ax b. Thus, 0 a ( x, y. Observe, the or reresents only ostve ntegers or only negatve ntegers, deendng on the sgn o a. In artcular, 0 and a 0 then ( x, s ostve dente. Moreover, b ac les that b (od ). Thus we have 0 (od ) or (od ), deendng on whether b s even or odd. Moreover, we wll wrte ( Z congruence classes whch are relatvely re to. Z / ) to denote the ultlcatve grou o We say that an nteger a s a quadratc resdue o x a (od ) has a soluton. When dscussng quadratc resdues, t s convenent to use Legendre sybols. I s an odd re and a an nteger relatvely re to, then ollows: a s dened as a x a (od ) has a soluton DEFINITION: otherwse Ths notaton allows us to concsely state soe well-known acts about quadratc resdues; here, q are dstnct odd res: ) ( ) ( ) / ) ( ) ( ) /8

4 TMME, vol6, nos.&,.39 q ( )( q) / a b ab ) ( ) v). q Ite () s called the Quadratc Recrocty Law; dscovered ndeendently by Euler and Legendre, the rst correct roo aeared n Gauss Dsqustones. Ites () and () are known as the Frst and Second Suleents to Quadratc Recrocty and were roved by Euler (79) and Legendre (785) resectvely. More generally, let sybol s dened as a k, and let a be any ostve nteger. The Jacob a a a k. Observe that a s a quadratc resdue a odulo, then, but the converse s not true. The Jacob sybol has any o the sae basc roertes as the Legendre sybol; n artcular the our results above are vald when and q are relaced by arbtrary odd ntegers. The Jacob sybol also a a a satses. The recrocty law or Jacob sybols was also roved by n n Gauss [7, Art 33], and can be stated as ollows: I and n are odd ntegers, then n n n ether o, n (od ) and n 3 (od ). n HISTORICAL BACKGROUND: The earlest nvestgatons concernng the reresentaton o ntegers by bnary quadratc ors were due to Ferat. In corresondence to Pascal and Marsenne, he claed to have roved the ollowng: THEOREM :. Every re nuber o the or k + can be reresented by x y.. Every re nuber o the or 3k + can be reresented by x 3y. 3. Every re nuber o the or 8k + or 8k + 3 can be reresented by x y. These results otvated uch later research on arthetc quadratc ors by Euler and Lagrange. Begnnng n 730, Euler set out to rove Ferat s results; he succeeded n rovng () n 79 (as well as the ore general Two-Square Theore), and ade sgncant rogress on the other two []. In a 7 aer ttled Theoreata crca

5 Bentea & Khosravan dvsors nueroru n hac ora aa qbb contentoru, Euler recorded any exales and orulated any slar conjectures (resented as theores). It was n ths aer that he also establshed any basc acts about quadratc resdues. Hs ost general result along these lnes was the ollowng: THEOREM : Let n be a nonzero nteger, and let be an odd re relatvely re to n. n Then x ny, gcd( x,. In 773, Lagrange ublshed the landark aer Recherches d arthetque, n whch he succeeded n rovng Ferat s conjectures concernng res reresented by the ors x y and x 3y. The sae aer contans a general develoent o the theory o bnary quadratc ors, treatng ors o the tye ax bxy cy. Lagrange s develoent o the theory s systeatc and rgorous t s here that he ntroduces the crucal concets o dscrnant, equvalence, and reducton. One o the rst results s a connecton between quadratc resdues and the reresentaton roble or general quadratc ors: THEOREM 3: Let be a natural nuber that s reresented by the or Then b ac s a quadratc resdue odulo. ax bxy cy. One o Lagrange s rary nnovatons was the concet o equvalence o ors (although the ternology s due to Gauss). We say that two ors are equvalent one can be transored nto the other by an nvertble ntegral lnear substtuton o varables. That s, and g are equvalent there are ntegers, q, r, and s such that ( x, g( x qy, rx s and s qr. It can be shown (e.g. see [6] or []) that equvalence o ors s ndeed an equvalence relaton. Moreover, equvalent ors have the sae dscrnant and reresent the sae ntegers (the sae s true or roer reresentaton). Gauss later rened ths dea by ntroducng the noton o roer equvalence. An equvalence s a roer equvalence s qr, and t s an roer equvalence s qr. Followng Gauss, we wll say that two ors are n the sae class they are roerly equvalent. Usng these deas, we obtan the ollowng

6 TMME, vol6, nos.&,. artal converse o Theore 3: THEOREM : Let be an odd re. Then s reresented by a or o dscrnant and only. Proo: Let ax bxy cy reresent, say ar brs cs. Because s re, we ust have gcd(r, s) =. Hence, we can wrte ru st or ntegers t, u. I g( x, ( rx ty, sx u, then g s roerly equvalent to and thus has dscrnant Moreover, by drect calculaton we have g x bxy cy. Thus, b c and so b (od ). Next, suose that (od ). We can assue that has the sae arty as (relacng by + necessar. Wrtng k, and recallng that 0 or (od ), we have k 0 (od ). Thus the or x xy ( k / ) y nteger coecents and reresents has Once we have arttoned the set o bnary quadratc ors nto equvalence classes, the next logcal ste s to choose an arorate reresentatve or each class. Ths naturally leads another o Lagrange s nnovatons, the concet o reducton. A rtve ostve dente or ax bxy cy s sad to be reduced b a c and b 0 ether b a or a c. Lagrange showed that every rtve ostve dente or s roerly equvalent to a unque reduced or, and that there are only there are only ntely any ostve dente ors wth a gven deternant. We wrte h () or the nuber o classes o rtve ostve dente ors o dscrnant. Thus, h() s the nuber o reduced ors o dscrnant. In the secal case where h ( n), the only reduced or o dscrnant -n n wll be the or x ny. In ths case, x ny. Ths stuaton s n act qute rare Gauss conjectured that the only values o n or whch h ( n) are n =,, 3,, and 7. The conjecture was roved by Landau n 903. More generally,

7 Bentea & Khosravan we call a undaental dscrnant t cannot be wrtten as k 0, where k > and 0 0 or (od ). Gauss conjectured that < 0 s a undaental dscrnant then h( = only or = -3, -, -7, -8, -, -9, -3, -67, -63. Ths was roved n 95 by Heegner []. GENUS THEORY: We say that two rtve ostve dente ors o dscrnant are n the sae genus they reresent the sae values n * ( Z / Z). Recall that equvalent ors reresent the sae ntegers and so ust be n the sae genus. Thus, the concet o genus rovdes a ethod o searatng reduced ors o the sae dscrnant accordng to congruence classes reresented by the ors. In hs table o reduced ors, Lagrange showed ors groued accordng to the congruence classes reresented by the ors. For ths reason, any authors credt the orgnal dea o genus to Lagrange. Soe authors have even attrbuted the dea to Euler [0]. However, Gauss s the rst to exlctly dscuss the concet o genus. More ortantly, he s the rst to ut t to use. Beore resentng Gauss denton o genus, a ew rearks concernng notaton and ternology are n order. Throughout ost o the Dsqustones Arthetca, Gauss assues ors have even ddle coecent that s, he ostly consders ors o tye ax bxy cy. (Fors wth odd ddle coecent are called roerly rtve, and are treated searately.) Instead o dscrnants, he uses the deternant o the or, dened as D b ac. Note that the dscrnant satses D. The ollowng result, ound n Artcle 9 o Dsqustones Arthetca, s the oundaton o genus theory. The roo s arahrased slghtly ro the orgnal text. THEOREM 5: Let F be a rtve or wth deternant D and a re nuber dvdng D: then the nubers not dvsble by whch can be reresented by the or F agree n that they are ether all quadratc resdues o, or they are all nonresdues. Proo: Let ag bgh ch and ag bgh ch. Then [ agg b( gh hg) chh] D( gh hg).

8 TMME, vol6, nos.&,.3 Thus s a quadratc resdue od D, and hence s also a quadratc resdue od or any dvdng D. It ollows that, are ether both resdues, or both are nonresdues od. That s, and are both reresented by F, then Fro the relaton D we get two ortant observatons: Frst, any odd re that dvdes D also dvdes. Moreover, s an odd re, then s a resdue od and only D s. Thus Theore 5 stll holds the word deternant s relaced by dscrnant. Henceorth, we wll revert to the ore coon ractce o usng dscrnants. The arguent used to rove Theore 5 also shows that 8 D or D, then the roduct o two nubers reresented by F wll be a quadratc resdue od 8 or a quadratc resdue od, resectvely. Hence 8 D, then exactly one o the ollowng s true: all nubers reresented by F are (od 8), or all are 3 (od 8), or all are 5 (od 8), or all are 7 (od 8). Lkewse, D, but 8 D, then all nubers reresented by F are (od ), or all are 3 (od ). These observatons are then used to classy ors accordng to characters. Let,,..., k be the odd re dvsors o D. Dene R the nubers reresented by F are quadratc resdues o, and N the nubers reresented by F are quadratc non-resdues o. We dene one addtonal character, 0, whch wll be an ordered ar a, b chosen ro the lst {(,), (3,), (,8), (3,8), (5,8), (7,8)}, where all nubers reresented by the or satsy a (od b). For exale, we wrte 0 =, to ndcate that all nubers reresented by the or are congruent to od. Fnally, the colete character or a or s then dened as:,..., 0 ;, k. Two ors then sad to be n the sae genus they have the sae colete character. In Artcle 3, Gauss dscusses the ossbltes or 0 based on the re actorzaton o the deternant, as well as the nuber o otental colete characters n each case. In each case, the nuber o otental colete characters s a ower o. Let n the table below: k be all o the odd res dvdng. We suarze the results

9 Bentea & Khosravan Possble 0 Nuber o otental colete characters r 8 (r > 0) k,8 3,8 5,8 7,8 k k, 3, k (od ), k k EXAMPLE: Let 55 ; then 0 =, and there are our reduced ors: Table 3 x x xy y xy 7 y,, x x xy 7 y 3xy y reresents, and s a resdue or any re, so the colete character or s,; R5, R. and 3 each reresent, whch s a non-resdue od 5 and od, so the colete character or each o these ors s,; N 5, N. Fnally, reresents, whch s a resdue odulo any odd re. Thus the colete character or s R5, R. It ollows that there are two genera, each wth two roer equvalence classes: Colete Character Reduced Fors,; R5, R x xy y, x 3xy y,; N 5, N x xy 7 y, 3 x xy 7 y Note that, 3 are equvalent, so they ust be n the sae genus. However, they are not roerly equvalent snce x, ). Thus they reresent two dstnct eleents ( 3 y wthn the genus. Observe also that n the exale above, there were our ossble colete characters, but only two actually dened a genus. In Artcles 6 and 87, Gauss

10 TMME, vol6, nos.&,.5 shows that the nuber o genera s always exactly hal the nuber o ossble colete characters and ust always be a ower o. For odd, non-square dscrnants, ths s easy to see: Let be an odd nteger reresented by a or o odd dscrnant, and let be an odd re dvdng. I R s a character, then, whereas N s a character, then. Relacng the characters by ther resectve Legendre sybols and ultlyng, we get k, where s the Jacob ( )( / k. By recrocty we have ) ) sybol and (. Snce s odd and (od ), we have. Fnally, snce s reresented by, we have by Theore 3. Thus, or reresented by, the roduct o the characters s always ; k o the characters are known, the k-th s also deterned. It ollows that there ust be k colete characters. Recrocty lays a crtcal role n the arguent above, and ths s no accdent. In Artcle 6, Gauss shows that at least hal the ossble colete characters cannot belong to a genus ths act serves as the bass o hs second roo o the Quadratc Recrocty [7, Art 6]. The arguent above (or Theore 3) shows that s reresented by a or o odd dscrnant, then. Gauss Theore 5 then allows us to extend ths relatonsh to eleents o ( Z / Z). That s, () s a well-dened a ro ( Z / Z) to {+}. Ths s a hooorhs snce. Moreover, ths s n n the unque hooorhs : ( Z / Z) { } such that q ker() and only q s reresented by a or o dscrnant A aous result o Drchlet guarantees that there are nntely any res n an arthetc rogresson, rovded the rst ter and

11 Bentea & Khosravan coon derence are relatvely re. Thus, each eleent o ( Z / Z ) * can be reresented as q, or soe odd re q not dvdng. Fro ths, t ollows that the the condton (q) or odd res q deternes unquely. q Let 0, (od ) be a dscrnant. The rncal or s dened by x y x xy y 0 (od ) (od ) The class and genus contanng the rncal or are called the rncal class and rncal genus, resectvely. reduced. When n, the rncal or s Note that the rncal or has dscrnant and s x ny. Many undaental roertes o genus can be descrbed n ters o the hooorhs and the rncal or: THEOREM 6: Gven a negatve nteger 0, (od ), let be the hooorhs o Theore, and let be a or o dscrnant. ) For an odd re not dvdng, ker() and only s reresented by one o the h () ors o dscrnant. ) ker( ) s a subgrou o ndex n ( Z / Z) ) The values n ( Z / Z) reresented by the rncal or o dscrnant or a subgrou H ker() v) The values n ( Z / Z) reresented by ( x, or a coset o H n ker( ). v) For odd, H { x x ( Z / Z) *} Part () o the theore s a restateent o Theore 3: ( ) = and only s reresented by soe or o dscrnant. Part () states that exactly hal the congruence classes n ( Z Z / ) are reresented by soe or o dscrnant ; or odd, ths ollows ro our arguent that exactly hal o all ossble colete characters actually result n a genus. Parts () and (v) get to the heart o genus theory; snce * dstnct cosets are dsjont, derent genera reresent dsjont classes n ( Z / Z). That s, we can now descrbe genera n ters o cosets kh o H n Ker (). We could then

12 TMME, vol6, nos.&,.7 dene a genus to consst o all ors o dscrnant that reresent the values o kh od. Note that ths denton could be used to show that each genus contans the sae nuber o classes [9, Art. 5]. EXAMPLE: Recall that there were our reduced ors o dscrnant = - 55: 3 x x xy y xy 7 y,, x x xy 7 y 3xy y There are ( 55) 55( )( ) 0 eleents n ( Z / 55Z). O these 0 eleents, 5 exactly 0 are reresented by a or o dscrnant -55. Snce ( x,0) x, the rncal or H Thus the set o classes n to be a subgrou o y {,, 9,,6, 6,3,3,36, 9} ( Z ( Z x xy reresents all o the squares: Z / 55 ) reresented by Z / 55 ). Also note that, s H, whch s easly vered ( 0, 7 y, so the set o classes reresented by, 3 can be wrtten as 7H {,7,8,3,7,8, 8,3, 3,5 }. O secal nterest are those dscrnants such that each genus contans exactly one class; n ths stuaton, the res that are reresented by a or o dscrnant are deterned by congruence condtons od See [] or detals COMPOSITION OF FORMS: The theory o cooston s ntrcately lnked to that o genus. Cooston o ors was rst nvestgated by Legendre and Lagrange, but the theory was brought to ruton by Gauss, who dscovered a rearkable grou structure. Gauss exoston s long and techncal, and s one o the ost dcult arts o the Dsqustones. However, the an result that classes o bnary quadratc ors o xed dscrnant or an abelan grou under the oeraton o cooston s justly celebrated as one o the lestones o 9 th century atheatcs. Matheatcans ollowng Gauss were able to strealne the theory consderably. Gauss showed that any two ors o the sae dscrnant can be coosed n such a way that cooston s a well-dened oeraton on (roer) equvalence classes o ors. For slcty, we resent a verson o the oeraton develoed by Drchlet [,

13 Bentea & Khosravan 3] whch s based on a case sngled out by Gauss or secal consderaton [7, Art ]. We say that ax b xy c y and a x b xy c y are concordant (the ternology s due to Dedeknd [3]) the ollowng condtons hold: ) a a 0 ) b b ) a c and a c I two concordant ors have the sae dscrnant, say b ac b ac, then ac ac, and so / a c / a c. We then dene the cooston o two concordant ors, o dscrnant as aa x bxy cy, where b b b and c c / a. Drchlet showed that gven two equvalence classes o ors C, a c / C, t s always ossble to nd concordant ors, wth C and C. Suose that ax bx y acy and ax bx y acy concordant ors. Then settng X xx cy y and Y ax y a yx by y, we ) have ( a x bx y a cy )( a x bx y a cy a a X bxy cy (by drect calculaton). Usng ths dentty and the denton o cooston gven above, we quckly deduce that reresents whenever reresents and reresents. The ollowng theore suarzes the an roertes o cooston [7, Art ]: are THEOREM 8 [Gauss]: For a xed dscrnant the set o equvalence classes o rtve ostve dente ors corse an abelan grou under the oeraton o cooston. The dentty o ths grou s the class contanng the rncal or. The class contanng the or ax bxy cy and the class contanng ts ooste ax bxy cy are nverses. Ths grou s called the class grou, and has cardnalty h(). The roo s long and techncal, as ght be exected; the results theselves reresent an unrecedented level o abstracton or ther te. Soon ater dscussng cooston o classes, Gauss denes dulcaton: let K and L be roer equvalence classes o ors o dscrnant D. I K K L, then we say that L s obtaned by dulcaton o K. In Artcle 7, Gauss onts out that the dulcaton o any class les n the rncal genus; n Artcles he shows the converse, statng that

14 TMME, vol6, nos.&,.9 t s clear that any roerly rtve class o bnary ors belongng to the rncal genus can be derved ro the dulcaton o soe roerly rtve class o the sae deternant. Ths act s oten reerred to n the lterature as the Prncal Genus Theore. Whle the stateent s ade rather casually (not even stated as a oral theore), Gauss nonetheless descrbes t as aong the ost beautul n the theory o bnary ors. (See [] or a dscusson o the any generalzatons o ths result.) We conclude wth a descrton o Gauss roo o the Prncal Genus Theore. To deonstrate how dulcaton o any class s n the rncal genus, Gauss denes cooston o genera, and n dong so descrbes another grou structure. In Artcle 6, he shows that, are rtve ors ro one genus, and g, g are rtve ors ro another genus, then the coostons g and g wll be n the sae genus. He then exlans how one can deterne the genus o g usng the characters or, g resectvely. Frst, he gves a ultlcaton table or the characters 0 ; then he descrbes ultlcaton o characters, as R and as N. The characters o g are then the roducts o,, 0,,..., k. I the dscrnant s odd, we can llustrate ths by relacng the characters by ther resectve Legendre sybols. Let k be odd, and let, g coe ro the genera G,G resectvely. Suose that s reresented by and that n s reresented by g, so the total characters o the ors can be descrbed as,,, k and n, n,, n k resectvely. Then G G s the genus wth total character n n n,,,. Note that the rncal genus always reresents, whch s a k quadratc resdue odulo any re; that s, or all. Thus the rncal genus G s the genus n whch all the characters have value. On the other hand, G s any

15 Bentea & Khosravan other genus and s an nteger reresented by G, the characters or G G wll be,,, =,,,. Hence G G G. Moreover, t ollows that the k genera or a grou o order, whose dentty s the rncal genus. BIBLIOGRAPHY. M. Bentea and A. Khosravan, Bnary Quadratc Fors: A Hstorcal Vew, Matheatcs and Couter Educaton, 0 (006), D.A. Cox, Pres o the For x ny, Wley-Interscence, John Wley and Sons, New York, L.E. Dckson, Theory o Nubers Vol. III: Quadratc and Hgher Fors, Chelsea, New York, 95.. L. Euler, Oeuvres Vol II, Gauther-Vllars and Sons, Pars, P. de Ferat, Oeuvres Vol II, Gauther-Vllars and Sons, Pars, D. Flath, Introducton to Nuber Theory, Wley, New York, C.F. Gauss, Dsqustones Arthetcae, Srnger-Verlag, New York, J.L. Lagrange, Recherches d arthetque, Oeuvres III, Gauther-Vllars, Pars, A.M. Legendre, Theore des Nobres, Pars, 830; rernt, Blanchard, Pars, F. Leereyer: The Develoent o the Prncal Genus Theore, ArXv Matheatcs e-rnts, ath/007306, 00.. W. Scharlau and H. Oolka, Fro Ferat to Mnkowsk, Lectures on the Theory o Nubers and Its Hstorcal Develoent, Srnger-Verlag, New York, J.P. Serre, b ac, Math. Medley 3 (985),. -0.