RECIPROCITY LAWS JEREMY BOOHER

Transcription

1 RECIPROCITY LAWS JEREMY BOOHER 1 Introduction The law of uadratic recirocity gives a beautiful descrition of which rimes are suares modulo Secial cases of this law going back to Fermat, and Euler and Legendre conjectured it, but the first comlete roof is due to Gauss, who in fact gave eight roofs It states that if and are distinct odd rimes, and at least one of them is congruent to 1 modulo 4, then is a suare modulo if and only if is a suare modulo If both are 3 modulo 4, is a suare modulo if and only if is not a suare modulo This is more clearly stated using the Legendre symbol Definition 1 Let be a rime and a be an integer relatively rime to Then a is defined to be 1 if x 2 a mod has a solution, 1 otherwise By convention, if a is a multile of the Legendre symbol is defined to be zero notation, the law can be comactly stated as follows Theorem 2 Quadratic Recirocity Let and be distinct odd rimes Then With this Two roofs relying on the same idea are given in Section 2: they are one of the roofs due to Gauss and a modern reformulation in terms of the language of algebraic number theory The search for generalizations of uadratic recirocity was a major goal in the develoment of algebraic number theory Versions for third and fourth owers were investigated by Gauss and Jacobi, and roven by Eisenstein Some of these generalizations are discussed in Section 3 A generalization and unification of many such laws took lace in the early twentieth century with the develoment of class field theory Section 4 briefly outlines how class field theory connects with the law of uadratic recirocity A great deal of information about the develoment of recirocity laws and many unusual roofs are to be found in Lemmermeyer [3] 11 Preliminary Facts The law of uadratic recirocity can be reformulated in terms of := 1 1 2, which has the roerty that 1 mod 4 This form matches the roofs we will give Theorem 3 Quadratic Recirocity Let and be odd rimes, and = Then = The euivalence relies on some standard roerties of the Legendre symbol Proosition 4 Let be a rime and a Z 1 a a 1 2 mod this is known as Euler s criterion 2 The Legendre symbol is a homomorhism from Z/Z to {±1} 3 = = Date: January 14,

2 2 JEREMY BOOHER 4 2 = The last two statements are known as the sulemental laws, since they deal with excetional cases not covered by the statement of uadratic recirocity These easy statements, as well as the euivalence of the two statements of uadratic recirocity, are roven in most elementary number theory books, for examle Ireland and Rosen [2, Chater 5] 2 Gauss s Proof and Algebraic Number Theory After introducing the slitting of rimes and the Frobenius element in basic algebraic number theory, a simle roof of the uadratic recirocity law becomes ossible Although the full language of algebraic number theory did not develo until the end of the 19th century, this roof is essentially due to Gauss After resenting the modern roof, we exlain how Gauss would have roven it using Gauss sums to byass non-existent language and techniues 21 Proof Using Galois Theory and Algebraic Number Theory To rove the main case of uadratic recirocity, Theorem 3, we will look at the slitting of rimes in the uniue uadratic subfield of a cyclotomic field Let and be odd rimes, and let be as before Consider the cyclotomic field Qζ, where ζ is a rimitive th root of unity Proosition 5 There is a uniue uadratic subfield K = Q of Qζ Proof We know that Qζ is a Galois extension of Q with Galois grou Z/Z As GalQζ /Q is cyclic, there is a uniue index two subgrou, which consists of the uadratic residues modulo By the fundamental theorem of Galois theory there is a uniue uadratic field K contained in Qζ and GalQζ /K is the subgrou of uadratic residues Furthermore, the discriminant of Qζ is a ower of, so the only rime that ramifies in K is But the only uadratic field ramified only at is Q 1 Now we wish to understand how the rational rime slits in the ring of integers of K There are two ways to do this: directly using an understanding of uadratic fields, and indirectly using the Frobenius of Proosition 6 The following are euivalent: 1 The rime slits in O K 2 The olynomial x 2 x factors modulo, ie 3 The element Frob GalQζ /Q fixes K = 1 Proof Standard algebraic number theory gives that 1 is euivalent to 2 Because 1 mod 4, the ring of integers O K is Z[ 1+ 2 ], so slits if and only if the minimal olynomial of x 2 x 1 4 factors modulo But this factors over F if and only if the roots lie in F, in other words = 1 Therefore the first two conditions are euivalent An argument using the Frobenius element shows that 1 and 3 are euivalent Recall that the Frobenius element Frob GalQζ /Q is a lift of the Frobenius automorhism of the residue field More recisely, let be a rime above in Z[ζ ], the ring of integers of Qζ, and let k be the residue field Z[ζ ]/, a finite extension of k = F Then Frob is an element of GalQζ /Q that reduces to the th ower ma in Galk /k It is uniue because the extension is Abelian and unramified at Likewise, we can define a Frobenius Frob for K over Q Looking at the reduction on residue fields, it is clear that Frob K = Frob Now the order of a Frobenius is the degree of the residue field extension, so slits comletely in O K if and only if Frob is the identity This haens if and only if Frob fixes K 1 The field Q is also ramified at 2

3 RECIPROCITY LAWS 3 The final ste is to obtain a criterion for when Frob fixes K using Galois theory Proosition 7 The Frobenius element Frob fixes K if and only if = 1 Proof Recall the isomorhism GalQζ /Q Z/Z identifies a Z/Z with the automorhism σ a over Q sending ζ to ζ a Now the residue extension k /k is generated by adjoining th roots of unity to k, so σ reduces to the Frobenius automorhism of the residue fields Since in this case the Frobenius is uniue, we see that Frob = σ Then Frob fixes K if and only if Z/Z lies in GalQζ /K, which by construction is the subgrou of uadratic residues modulo Combining Proositions 6 and 7 gives that = = 1 if and only if = 1, so This comletes the roof of the law of uadratic recirocity in the guise of Theorem 3 22 Proof Using Gauss Sums The main idea in the alternative formulation is that Gauss sums allow us to construct K and identify the slitting of concretely We first recall their definition and basic roerties, then rehrase the above roof in Gauss s language Let n be a ositive integer and χ be a grou homomorhism Z/nZ C Since all irreducible reresentations of Z/nZ are one dimensional, χ is often simly called a character It can be extended to a ma Z C by eriodicity and setting it to be 0 on integers not relatively rime to n Fix a rimitive nth root of unity ζ n Definition 8 The Gauss sum gχ is defined to be gχ = χxζn x x Z/nZ Remark 9 The function x ζn x is an additive character of Z/nZ In general, a Gauss sum is a combination of a multilicative character with an additive character A fruitful analogy is the grou R + and the Gamma function If is an odd rime, let χ denote the uadratic character, so χ a = a We are mainly interested in gχ, and would like to reduce it modulo This is a comlex number so a riori this makes no sense However, the Gauss sum can be interreted as an algebraic integer or an element of a finite field as follows Since χ takes on only the values ±1, gχ lies in Z[ζ ], the ring of integers of Qζ It is therefore ossible to reduce this modulo Alternately, we can interret the definition as an element in the finite extension of F obtained by adjoining th roots unity We can use Gauss sums to recover the uniue uadratic subfield of Qζ by calculating the size of gχ Then the algebraic number theory in the roof can be relaced by concrete calculations with the Gauss sum The following roosition is analogous to Proosition 5 Proosition 10 For a rime, gχ 2 = Thus Qgχ is a uadratic subfield of Qζ Proof To rove this, we will use a twisted Gauss sum g a χ = x F χ xζ ax Note that g 0 χ = 0 since half the elements of F are uadratic non-residues and half are uadratic residues We will calculate S = g a χ 2 = g a χ 2 a F a F

4 4 JEREMY BOOHER in two different ways On one hand, as multilying by a 0 is a ermutation of F so g a χ = χ a 1 χ axζ ax = χ a 1 gχ ax F Thus we can evaluate S as S = a F g a χ 2 = On the other hand, a F χ a 2 gχ 2 = gχ 2 a F S = χ xχ yζ ax+y a F x,y F = χ xχ y ζ ax+y x,y F a F 1 = 1gχ 2 When x + y 0, the inner sum is a sum over all th roots of unity, so euals 0 Otherwise it is the sum of ones, and χ xχ x = Therefore we obtain S = = 1 x F Euating the two exressions for S gives gχ 2 = Remark 11 The roblem of determining the sign of gχ is more delicate - see for examle Ireland and Rosen [2, 64] It is related to the functional euation for Dirichlet L-functions Let us try to use this alternative descrition of this uadratic subfield to determine when a rime is slit without using this language We have the following, analogous to Proosition 6 Proosition 12 Let be an odd rime not eual to Then = 1 if and only if gχ gχ mod Proof Recall that Euler s criterion says that a a 1 2 mod So using Proosition 10, gχ mod Multilying by gχ, we see the euivalence Of course, the condition that gχ = gχ mod is exactly the condition, suitably interreted, that Frob fixes the uadratic subfield Qgχ Finally, we obtain an analogue of Proosition 7 by a direct calculation Proosition 13 Let be an odd rime not eual to Then gχ = gχ mod Proof Recall that a + b a + b mod, so gχ a a F a F ζ a a ζ a 1 a F mod a ζ a mod mod

5 RECIPROCITY LAWS 5 where the last ste uses that a is ±1, so raising it to an odd ower does not change it But the last sum is just gχ since multilying by is a ermutation of F Therefore gχ gχ mod Combining the last two roositions, we see that uadratic recirocity = 1 if and only if = 1, again yielding 3 Generalizations of Quadratic Recirocity There are two obvious directions to generalize uadratic recirocity: ask the same uestions about nth owers instead of suares, and ask the uestion about the residue fields of number fields other than Q To get nice answers, one needs the nth roots of unity to lie in the number field 31 Cubic Recirocity The simlest examle of this is the law of cubic recirocity, which addresses the uestion of third owers in Qζ 3 The first ste is to define a cubic residue symbol using the idea behind Euler s criterion Definition 14 For α Z[ζ 3 ] and π a rime of Z[ζ 3 ], define the cubic residue character α π the uniue root of unity congruent to α Nπ 1/3 modulo π If π 3, define α π = 0 3 to be Just as we needed to use instead of to obtain a nice formulation of uadratic recirocity, we need a way to distinguish one of the six associates of a rime in Z[ζ 3 ] Definition 15 If π is a rime in Z[ζ 3 ], then π is rimary if π 2 mod 3 The main law of cubic recirocity is now easy to state Theorem 16 Cubic Recirocity Let π 1 and π 2 be distinct rimary rimes not above 3 Then π1 π2 = π 2 3 There are also sulemental laws for the rime above 3 and the units The first roof, first ublished by Eisenstein in 1844, uses the same techniues aearing in the revious roofs of uadratic recirocity The necessary result about Gauss sums is the following analogue of Proosition 10 Proosition 17 Let π be a rimary rime and χ π be the cubic residue character Then gχ π 3 = π 2 π The elementary roof of this uses Jacobi sums, which for characters χ 1 and χ 2 is Jχ 1, χ 2 := x F χ 1 xχ 2 1 x A comlete roof is contained in Ireland and Rosen [2, Chater 8 and 9] The roof of cubic recirocity roceeds in cases, deending on whether π 1 and π 2 are inert or slit rimes For examle, suose π 1 = is inert and π 2 slit, with norm Then the roosition combined with the definition of the cubic residue character tells us that π 1 3 gχ π2 2 1 χ π 2 mod Evaluating gχ π2 2 directly using the binomial theorem and then comaring gives one case of cubic recirocity The other cases are similar

6 6 JEREMY BOOHER 32 Eisenstein Recirocity The Eisenstein Recirocity Law generalizes uadratic and cubic recirocity to deal with th owers in the cyclotomic field Qζ 2 Two of the difficulties are the failure of uniue factorization, necessitating the use of ideals, and the more comlicated slitting of the Gauss sums The ower residue symbol is a generalization of the uadratic and cubic character Let be a rime, and a rime ideal in O Qζ above Suose that is relatively rime to Definition 18 Let α Z[ζ ] and be a rime ideal of Z[ζ ] not containing The th ower residue symbol, α is defined, if α, to be the uniue th root of unity such that αn 1/ α mod If α, define the residue symbol to be 0 Like the Legendre symbol and cubic residue character, the ower residue symbol is obviously multilicative and deends on α only modulo As in the case for cubic recirocity, we need to deal with the ambiguity introduced by units Definition 19 A nonzero element α Z[ζ ] is called rimary if it is not a unit, is rime to, and is congruent to a rational integer modulo 1 ζ 2 We can now state the th ower recirocity law Theorem 20 Eisenstein Recirocity Let be an odd rime, a Z rime to, and α Z[ζ ] a rimary element Suose furthermore α and a are corime Then α a = a α The roof of this theorem is found in Ireland and Rosen [2, Chater 14] A major ingredient is the the Stickelberger Relation, which gives the non-obvious factorization for Gauss sums in Qζ : it is a generalization of Proosition 10 and 17 4 Class Field Theory and Recirocity Laws All of the above recirocity laws can be roven using class field theory, in articular the existence of the Artin ma identifying the Galois grou of an Abelian extension of number fields with a uotient of the idele class grou 3 There are several ways to roceed outlined in Exercises 1 and 2 of Cassels and Frölich [1]: we will use Hilbert symbols and the Hilbert recirocity law Let n a ositive integer, K be a number field containing the nth roots of unity, and v a lace of K Let a, b K One can define the norm residue symbol a, b v using the local Artin ma ψ v for the extension K n a/k via the formula n a ψvb = a, b v n a This turns out to be a bilinear ma K K µ n that can be studied using class field theory One imortant roerty is the roduct formula Theorem 21 Let a, b K Then v a, b v = 1, where the roduct ranges over all laces of K Given class field theory, this is not too difficult to rove In the case that n = 2, the norm residue symbol has a much more exlicit descrition, and is known as the Hilbert symbol 2 Additional secial cases were roven first, most notably biuadratic recirocity which deals with fourth owers in Z[i] 3 This is often called the Artin recirocity law, not because it looks like other recirocity laws but because secial cases give many of the recirocity laws

7 RECIPROCITY LAWS 7 Lemma 22 The norm residue symbol a, b v is 1 if and only if the euation z 2 ax 2 by 2 = 0 has a non-trivial solution in O Kv The roof of this lemma reuires local class field theory Note that the euation having a nontrivial solution is euivalent to saying that a is a norm from the extension K b/k In this secial case, the roduct formula is known as the Hilbert recirocity law When K = Q, it is euivalent to the law of uadratic recirocity We will show how to deduce uadratic recirocity from it by erforming local calculations for the Hilbert symbols Let and be distinct odd rimes, and = as before We first consider the case when v is a finite lace of Q that is not eual to or Proosition 23 If v,,, then, v = 1 Proof We first treat the case when v 2 We need to show that z 2 x 2 y 2 = 0 has a nontrivial solution in Q v Chevalley s theorem says that the number of solutions in F v is congruent to zero modulo v Chater 1 Theorem 3 of Serre [4], so since 0, 0, 0 is a solution there must be a non-zero one as well Since v 2, Hensel s lemma allows us to lift it to Q v When v = 2, Hensel s lemma reuires a solution modulo 8 to lift to Q 2 Since 1 mod 4, there are two cases: if 1 mod 8, take z x 1 mod 8, y 0 mod 8 if 5 mod 8, take z x 1 mod 8 and y 2 mod 8 Then aly Hensel s lemma Now we deal with the case that v = or Since a, b v = b, a v, it suffices to consider v = Proosition 24 If v =, then, = Proof Suose z 2 x 2 y 2 = 0 has a non-trivial solution in Q This is homogenous, so we may assume the solution is in Z and that one of the variables is relatively rime to If z, then since z 2 x 2 y 2 = 0 we see that y and hence x 2 = z 2 y 2 0 mod 2 Thus x, a contradiction Likewise y Reducing modulo, we see that z 2 y 2 mod, so is a suare modulo Therefore if, = 1, then we have = 1 Conversely, suose = 1, so by Hensel s lemma there is an α Z such that α 2 = Then α = 0, so z 2 x 2 y 2 = 0 has a non-trivial solution Finally we deal with the infinite lace Proosition 25 If v =, then, = 1 Proof The only obstruction to z 2 x 2 y 2 = 0 having a non-trivial solution in Q = R is the sign of and As long as x 2 +y 2 takes on a ositive value, a suare root exist But x 2 +y 2 can be made ositive since is always ositive We can now combine these calculations with the Hilbert recirocity law to deduce uadratic recirocity Theorem 21, combined with Proosition says that 1 =, v =,,, = v It is worth noting that Hilbert recirocity allows the sulemental laws to be recovered using the same techniues References 1 JWS Cassels and A Fröhlich, Algebraic number theory: Proceedings of an instructional conference organized by the london mathematical society a nato advanced study institute with the suort of the international mathematical union, London Mathematical Society, 2010

8 8 JEREMY BOOHER 2 K Ireland and M Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, Sringer, F Lemmermeyer, Recirocity laws: From euler to eisenstein, Sringer Monograhs in Mathematics, Sringer, JP Serre, A course in arithmetic, Graduate Texts in Mathematics, Sringer, 1973