Parvati Shastri. 2. Kummer Theory

Reciprocity Laws: Artin-Hilert Parvati Shastri 1. Introduction In this volume, the early reciprocity laws including the quadratic, the cuic have een presented in Adhikari s article [1]. Also, in that article we have seen an exposition on the Eisenstein reciprocity without the use of class field theory. In the International Congress of Mathematicians, 1900, Hilert asked for the most general reciprocity law, (Hilert s prolem 9) which would hold in any numer field. In order to formulate and prove such a general reciprocity law, Hilert introduced the norm residue symol known after him as the Hilert Symol, in place of the power residue symol and proved a reciprocity law for this symol. From this law, one can derive all the earlier power reciprocity laws. Later on, Artin introduced a symol named after him as the Artin Symol, and proved a reciprocity law for his symol. This is the crux of class field theory, today. In this article, we shall assume ut recall the relevent theorems from class field theory, and deduce Hilert s reciprocity law and show how this would imply the power reciprocity laws, that you have seen earlier. Further, we will also explicitly derive the quadratic reciprocity law, without the use of class field theory. We egin with Kummer Theory. 2. Kummer Theory Theorem 1 Let K e a field of characteristic 0, and µ n K, e the group of all n th roots of unity. 1 Let e a sugroup of K such that K n K and L = K( n ). Then L/K is a Galois extension (in fact, aelian of exponent n) and there exists a canonical isomorphism 2 K n Hom (G(L/K), µ n). Proof: (Sketch) Assume K /K n is finite. a. Define χ a : G µ n Let G = Gal(L/K) and let 1 This result holds also in characteristic p > 0 under the additional hypothesis that, (chark, n) = 1. 2 If K /K n is infinite, then in the case when /K n is also infinite, or equivalently, L/K is infinite, one needs to take continuous homomorphisms. However, we need to apply this result in the case when K /K n is finite, and we sketch a proof for this case only. 175

176 PARVATI SHASTRI y χ a (σ) = σ( n a) n a. We have a homomorphism θ : Hom(G, µ n ) defined y θ(a) = χ a. It is clear that ker θ = K n. We claim that θ is surjective. Let χ Hom(G, µ n ). Then y Dedekind s theorem on the linear independence of characters, the sum σ G χ(σ)σ 1 0. Let λ L e such that = σ G χ(σ)σ 1 (λ) 0. For τ G, we have, τ() = τ ( σ G χ(σ)σ 1) (λ) = σ G χ(σ)(τσ 1 )(λ) = χ(τ) ( σ G χ(τ 1 σ)τσ 1) (λ) = χ(τ). Let a = n. Then a K n and we get θ(a) = χ. Therefore we get an induced isomorphism, K n = Hom(G, µ n ). Remark 1 The aove correspondence gives a ijection etween sugroups of K with K n K and aelian extensions of exponent n. (For a proof of this statement as well as for generalization to infinite extensions, the reader can refer to [4], Chapter 8, or [3], p.15. See also [2], 4, this volume.) 3. Local Reciprocity Law Let K e a local field of characteristic 0. By this we mean, a finite extension of Q p. For any local field K, we fix the following notations. O K ring of integers of K m K the maximal ideal of K π K a generator for m K U K group of units of K κ(k) the residue class field O K /mo K. The following is the main theorem of local class field theory.

RECIPROCITY LAWS : ARTIN-HILBERT 177 Theorem 2 Let L K e a finite aelian extension of local fields with Galois group G(L K). Let N L K : L K denote the norm map. Then there exists a canonical isomorphism r L K : G(L K) K /N L K L We will not prove this theorem, ut riefly discuss how the isomorphism is defined. First assume that L K is unramified. Then r L K can e descried as follows. We know that K = U K (π K ), where (π K ) is the (multiplicative) cyclic group generated y π K. Since L K is unramified, N L K is surjective on the unit group, that is, N L K (U L ) = U K. Clearly N L K (π) = π n, where n = [L : K]. It follows that K /N L K (L ) = (π)/(π n ), since K n N L K (L ). On the other hand, if L K is unramified, the Galois group G(L K) is isomorphic to the Galois group of the residue classfield extension κ(l) κ(k). This is a finite extension of a finite field. Recall that a finite extension of a finite field is cyclic and there is a distinguished generator for the Galois group, viz., the Froenius. So there is a unique generator of G(L K) which corresponds to the Froenius. Let us denote this generator y φ. 3 The reciprocity map r L K is given y, r L K (φ) = π mod N L K (L ). In the general case, r L K is defined, suject to the following two properties: (i) (Functoriality) If L K and L K are finite Galois extensions of local fields with K K, L L, then the diagram G(L K ) r L K K /N L K L Res N K K G(L K) r L K K /N L K L is commutative, where Res denotes the restriction map Res(σ) = σ L σ G(L K ). (ii) If L K is a finite unramified extension, then r L K is simply the map r L K (φ L K ) = [π], where [π] is the class of π mod N L K (L ). Let L K e a totally ramified cyclic extension and σ e a generator for G(L K). Then one can show that there exists a finite aelian extension Σ 3 φ is characterized y the property, where q is the cardinality of κ(k). φ(x) x q mod π x O K,

178 PARVATI SHASTRI of K such that LΣ Σ is unramified and the restriction of the Froenius of LΣ Σ to L is σ. Then r L K (σ) = N Σ K (π Σ ). The general case is reduced to the cyclic case. Also this map is surjective and the kernel of this map is precisely the commutator sugroup [G, G]. Thus, there is a canonical (i.e., satisfying (i) and (ii) aove) isomorphism r L K : G(L K) a K /N L K (L ). In particular, for finite aelian extensions, the Galois group G(L K) is isomorphic to the norm residue group K /N L K (L ). 4. Local Artin Symol Let the notation e as in section 2. Let (, L K) e the inverse of the reciprocity map. By composing it with the natural map K N L K (L ), we get for every a K, a symol which we still denote y (a, L K) taking values in G(L K). This is called the Artin symol; i.e., the local Artin symol is induced y the inverse of the local reciprocity map. Oserve that we have the following simple description of the Artin symol in the special cases a = π, u where π is a parameter and u is a unit in K, viz., (π, L K) is the Froenius G(L K) and 5. Hilert Symol (u, L K) = 1. We now define the Hilert Symol. Let µ n e the group of n th roots of unity. Assume that K is a local field containing µ n. We have, y Kummer Theory, K /K n = Hom (G(L K), µn ), where L = K( n K ). (Note that K /K n is finite, since K is a local field.) On the other hand, local reciprocity law gives an isomorphism, K /N L K (L ) = G(L K). Since, K n N L K (L ) K, it follows that K n = N L K L. Hence we get a pairing,, n : K /K n K /K n µ n given y a, n = χ ((a, L K)),

RECIPROCITY LAWS : ARTIN-HILBERT 179 where χ is the character associated to y Kummer Theory, and (a, L K) is the local Artin symol. By the properties of the Artin symol, it follows that a, n = χ ((a, L K)) = (a, L K) ( n ( ) a, K( n ) ) K) ( n ) n = n µ n. By composing it with the natural map K K /K n, we get a pairing, n : K K µ n. This is called the Hilert symol of degree n. In what follows, we will fix an n, and drop the suffix n. Remark 2 It follows easily y the definition that the Hilert symol is non degenerate in the sense that, a, = 1 K a K n and a, = 1 a K K n. We now recall a few asic properties of the Hilert symol, which are needed in the sequel. Lemma 1 The Hilert symol has the following properties: (i) (Bimultiplicativity) aa, = a,. a,, a, = a,. a, a, K. (ii) 1 a, a = 1 = a, 1 a a K. (iii) a, 1 = a, 1 = a, a = 1 = a, 1. (iv) (Skew symmetry) a, =, a 1. Proof: Part (i) follows y definition (easy to check). For proving Parts (ii), (iii) and (iv), oserve that, for a, K, a, = 1 if and only if a is a norm from K( n ). We have, n ( 1 a = 1 ζ i n ) a, i=1 where ζ is a primitive n th root of unity, i.e., 1 a is a norm from K( n a). So (ii) follows.

180 PARVATI SHASTRI Next, y (i) we have, a, 1. a, 1 = a, 1. Hence a, 1 = 1. Similarly, a,. a, 1 = a, 1 = a, 1 = 1 Now, oserve that a = 1 a. 1 a 1 Therefore, if we take = a, we get a, a = 1. This completes the proof of Part (iii). (iv) By (iii) we have, a, a = 1. Now use imultiplicativity and simplify using (iii) to get (iv). 6. Power Residue Symol We now assume that (n, p) = 1 where p is the characteristic of the residue class field and compute the Hilert symols u, v and π, u, where u, v are units and π is a parameter of K. It follows from standard facts of local theory, that K( n v) is unramified, and that the norm function is surjective on the unit group. Therefore u, v = 1 units u, v K. Also, y the discussions in Section 4, we know that the parameter corresponds to the Froenius under the reciprocity map. Thus ( π, K( n u) K ) (x) x q mod π x O L, where L = K( n u). In particular, (π, K( n u) K) ( n u) n u q mod π u q 1 n. n u mod π. So π, u u q 1 n mod π. We define the n th power residue symol, y ( ) u = π, u. π Note that π, u is a root of unity in K and is independent of the parameter chosen. 7. Artin s Reciprocity Law Let K e a numer field. Let V K e the set of all valuations of K including the archimedian ones. Let L K e a finite aelian extension of K. For every valuation v V K, fix a valuation w of L, which extends v. Note that the archimedian completions of K v are isomorphic to either R or C. Therfore either L w = Kv = R or C or Lw = C is a quadratic extension of K v = R, with Galois group isomorphic to Z/2Z. In order to state Artin s reciprocity law, we need to define Artin s symol at the archimedian completions also. If L w K v is quadratic, let σ e the nontrivial automorphism of L w K v. We define (a, L w K v ) = 1, if a > 0, (a, L w K v ) = σ if a < 0. If L w = Kv we define (a, L w K v ) = 1. With this notation, we have,

RECIPROCITY LAWS : ARTIN-HILBERT 181 Theorem 3 For any a K, 8. Hilert s Reciprocity Law v V K (a, L w K v )) = 1. 4 As in the case of Artin symol, we also need to extend Hilert s symol at the archimedian completions, in an ovious manner. For a, R, define With this definition, we have sgn a 1 sgn 1 a, := ( 1) 2 2. Theorem 4 Let K e a numer field, µ n K, and V K e as in the previous section. Let a, K. Then v V K a, v = 1, where a, v is the n th Hilert symol at the completion K v. Proof: This is immediate from Artin s reciprocity law. In fact, we have, v V K a, v = ( v χ a, K v ( n ) ) = ( v (a,kv( n )))( n ) n = Id( n ) n = 1. (The last ut one equality is y Artin s reciprocity law.) 9. Power Reciprocity Law We need to define gloal power residue symol, in terms of the local symols. Let K e a numer field containing µ n, as in the previous section. Let a, K. Let () = p p n i e the factorization of the principal ideal (). Let vp e the valuation on K corresponding to the prime ideal p. We define the power residue symol ( ) a to e the product of the local power residue symols; i.e., ( ) a = p ( a ) 4 Note that this would also mean that all ut finitely many terms in this product are equal to 1. v p.

182 PARVATI SHASTRI This is well defined, since the local power residue symol is independent of the parameter chosen. Theorem 5 Let K e a numer field, µ n K, a, K. Let (a), (), (n) e relatively prime and let Then V n = {vp : p (n)} {v : v is archimedian}. ( ) ( ) a 1 = a, v. a v V n Proof: Let us look at Hilert s reciprocity law, v V K a, v = 1. Let V 0 = {v V K : v an } and V 1 = V K V 0. The left hand side can e written as a, v a, v a, v a, v. v V 1 Now, oserve that v v v n v V 1 a, v = 1, since each of the symols is trivial. Also, ( a, v = a) v if v a and ( ) a 1 a, v =, a 1 v = v Hence the theorem follows. 10. Quadratic Reciprocity Law if v. In order to derive the quadratic reciprocity law, we need to compute the Hilert symols in the special case, n = 2, K = Q. First assume a, are odd positive integers. Then we have, ( ) ( a = a, 2 a,. a) Here a, 2 denotes the Hilert symol at the dyadic completion Q 2. Since a, are positive, a, = 1. So, we only need to compute a, 2. Let U Q2

RECIPROCITY LAWS : ARTIN-HILBERT 183 e the unit group of Q 2. Note that U Q2 /U 2 Q 2 is generated y {5, 1}. 5 By the imultiplicative and skew symmetric properties of the Hilert symol, it is enough to compute 5, 1 2, 5, 5 2, 1, 1 2. We have, 5, 1 2 = 5, 5 2 = 1 and 1, 1 2 = 1. From this it follows that ( ) ( a. = ( 1) a) a 1 1 2 ( 1) 2, for a, {5, 1}. For aritrary odd integers, the result follows from the multiplicativity of these symols and the fact that for an odd integer a, a 2 1 mod 4. REFERENCES 1. S.D.Adhikari,The Early Reciprocity Laws: From Gauss to Eisenstein, This volume. 2. M. J. Narlikar, Aelian Kummer Theory, This volume. 3. J.Neukirch, Class Field Theory, Springer-Verlag, 1986. 4. V. Suresh, Chapter 8 in Introduction to Class Field Theory, (Lecture Notes of the Instructional School on Algeraic Numer Theory, held in the Department of Mathematics, University of Mumai, Decemer 1994-January 1995). 5. J. Tate, Prolem 9: The General Reciprocity Law, Proceedings of Symp. in Pure Math. Vol. 28, 1976, pp. 311-322. Parvati Shastri Department of Mathematics University of Mumai Mumai 400 098 e-mail: parvati@math.mu.ac.in 5 This is a consequence of Hensel s lemma.