Kummer Theory and Reciprocity Laws Peter Stevenhagen Abstract. Insert abstract here. 1. Introduction How can we find abelian extensions of a number field? For any such field K, we have the cyclotomic extension K ⊂ K(ζm); the Galois group will be abelian and a ∗ subgroup of (Z/m√ Z) . We might also adjoin a square root, but should we adjoin n a general root a, we no longer even have a Galois extension, unless ζn ∈ K already. But in fact, for K = Q, all quadratic extensions of Q are already cyclotomic: for p 6= 2, there is only one cyclotomic extension which is ramified only at p ∗ ∼ and only tamely ramified, namely, Q(ζp). This field has Galois group (Z/pZ√) = Z/(p−1)Z, so it has a unique quadratic subfield, which is easily seen to be Q( p∗), where p∗ = (−1)(p−1)/2p: it is the unique quadratic field ramified only at p. In √ √ √ addition, we have Q( −1, 2) = Q(ζ8), so we find Q( a) ⊂ Q(ζ4|a|). Already this discussion of quadratic subextensions of cyclotomic extensions gives us the classical quadratic reciprocity law. Given distinct odd primes p, q, then there is a Legendre symbol (p/q) which is 1 or −1, respectively, depending on whether or not p is a square modulo q or not, respectively. One has the quadratic reciprocity law where q , p or q ≡ 1 (mod 4), p p = q q − , else. p We can write this more compactly as p q = (−1)(p−1)(q−1)/4. q p To prove this, consider first the case q ≡ 1 (mod 4). We consider the cyclo- tomic extension (ζ )/ , with Galois group ( /q )∗. This extension contains the Q q √Q Z Z quadratic extension ( q). Let σ ∈ ( /q )∗ be the Frobenius at p, and note that Q √ √ Z Z (p/q) = 1 if and only if σp( q)/ q = 1. By the property of the Frobenius, we 2 Peter Stevenhagen have √ √ √ p−1 σp( q)/ q ≡ ( q) (mod p), so this is equal to 1 if and only if q(p−1)/2 ≡ 1 (mod p), which holds if and only if (q/p) = 1 by Euler’s criterion. This proves quadratic reciprocity. The case when √ q ≡ 3 (mod 4) is similar, except now we work in Q( −q), and we end up with an extra factor (−1)(p−1)/2. Seen one way, then, the quadratic reciprocity law is none other than the statement that the quadratic extensions are all contained in cyclotomic extensions, over which we have control. 2. Kummer Theory We would like to generalize the quadratic reciprocity law; to do so, we need to construct abelian extensions of number fields K. √ n Throughout we assume that ζn ∈ K. Then L = K( a) (the splitting field of Xn − a) is abelian over K, for we have a map Gal(L/K) ,→ hζni √ σ( n a) σ 7→ √ . n a √ Note that this is independent of the choice of n a (they differ by a root of unity, which since they are in K, drops out), and by a little work we see that in fact this is a group homomorphism. Kummer theory gives a certain converse to this statement: If L/K is cyclic of n order n, and ζn ∈ K, then there exists an α ∈ L such that L = K(α) and α ∈ K. This was basically discovered by Lagrange: If G = hσi, then for any x write down the Lagrange resultant −1 −2 2 1−n n−1 α = x + ζn σ(x) + ζn σ (x) + ··· + ζ σ (x). We see that σ(α) = ζnα, and if α 6= 0, then such an α will generate the extension. We can always find such an α 6= 0; this is a result from Galois theory (known as ‘Artin-Dedekind’ or linear independence of characters). More generally, Kummer theory tells us that if L/K is Galois with group n G which is abelian of√ exponent n (meaning that for all σ ∈ G, σ = idL), and n ∗ n ∗ ζn ∈ K, then L = K( W ) for some subgroup (K ) ⊂ W ⊂ K . Then Kummer theory tells us that (within a fixed algebraic closure) there is a bijection {W :(K∗)n ⊂ W ⊂ K∗} o / {L ⊃ K abelian, exponent n (inside K)} √ W / K( n W ) (L∗)n ∩ K∗ o L. Kummer Theory and Reciprocity Laws 3 In this case, if W ↔ L, then we have a perfect pairing ∗ n Gal(L/K) × W/(K ) → hζni √ σ( n a) (σ, a) 7→ √ ; n a that is to say, ∼ ∗ n Gal(L/K) = Hom(W/(K ) , hζni). If L/K is a cyclic extension of degree n but ζn 6∈ K, it can be very hard to describe. However, when one adjoins ζn to K, by√ Kummer theory, the extension n L(ζn) is now Kummer over K(ζn), so L = K(ζn, a) for some a ∈ K(ζn). Note that this extension is not abelian, but it is abelian in two steps, which is usually good enough. √ n L(ζn) = K(ζn, a) ppp ppp ppp ppp L p K(ζn) oo ooo ooo ooo K o 3. Norm Residue Symbol We now look at the local situation. Recall the statement of local class field theory: if F = Kp is a local field, and E ⊃ F is a finite abelian extension, then ∗ ∗ ∼ F /NE/F E −→ Gal(E/F ) and we have a bijection between open subgroups of F ∗ and finite abelian extensions of F . √By Kummer theory, if ζn ∈ F , then E ⊃ F of exponent n arises as E = F ( n W ) where (F ∗)n ⊂ W ⊂ F ∗. But by class field theory, E ⊃ F of exponent n ∗ n ∗ ∗ arises as (F ) ⊂ NE/F E ⊂ F . These two ways of viewing extensions are dual to each other. √ The maximal exponent n extension by Kummer theory is E = F ( n F ∗). This is a finite extension, for in the local case (F ∗)n has finite index in F ∗. Since the bijection in class field theory is inclusion-reversing, NE∗ = (F ∗)n. By the perfect pairing, we have ∗ ∗ n ∗ ∗ n F /(F ) × F /(F ) → hζni √ σ ( n β) (α, β) 7→ α√ n β This is called the nth power norm residue symbol, and we denote it F ,(−, −)n,F . It has two useful properties: 4 Peter Stevenhagen √ n ∗ • From local class field theory, (α, β)n,F = 1 if and only if α ∈ N(F ( β) ). ∗ ∗ ∼ • Since AF /NE/F AE −→ I, the inertia group, if the extension E/F is un- ∗ ramified and α ∈ AF , then (α, β)n,F = 1. ∗ ∗ Example 3.1. Take the case F = Qp, n = 2, p 6= 2. Then Qp = hpi × Zp, so ∗ 2 2 ∗ 2 ∗ 2 ∗ (Qp) = hp i × (Zp) , and so (Qp) has index 4 in Qp, where ∗ ∗ 2 ∼ ∼ Qp/(Qp) = hpi × hai = Z/2Z × Z/2Z, for a ∈ with (a/p) = −1. Z √ √ We have (x, y)2,Qp ) = σx( y)/ y. In the unramified case, y = a, we see that (a, a) = 1, and since πF 7→ FrobE/F , we see (p, a) = −1. In the ramified case, y = p the norms must generate a subgroup of index 2, so (a, p) = −1; and finally we √ ∗ √ wish to know if p ∈ N(Qp( p) ): but (p, p)(−1, p) = (−p, p) = 1 (N( p) = −p), (p−1)/2 so (p, p) = (−1, p) = (−1) . We summarize the values of (x, y)2,Qp in the following table: x \ y a −p a 1 −1 p −1 (−1)(p−1)/2 The symbol has other important properties: • (−α, α) = 1 for all α ∈ F ∗; • (1 − α, α) = 1 for α ∈ F ∗ \{1}; • (α, β) = (β, α)−1. The latter property, for example, follows from (−αβ, αβ) = (−α, α)(β, α)(α, β)(−β, β) = (β, α)(α, β) = 1. Note that we also have this pairing for archimedean F . For F = C this is trivial, for F = R we have the Artin isomorphism ∗ ∗ Gal(C/R) R /NC −−−−−−→ and a corresponding pairing ∗ ∗2 ∗ ∗2 R /R × R /R → h−1i with (−1, −1)∞ = −1. Now, for a number field K such that ζn ∈ K, for any prime p of K, there is a norm residue symbol ∗ ∗ n ∗ ∗ n (−, −)n,p : Kp /(Kp ) × Kp /(Kp ) → hζni. We then have: Proposition 3.2 (Product formula). For α, β ∈ K∗, Y (α, β)n,p = 1. p≤∞ Kummer Theory and Reciprocity Laws 5 Example 3.3. To complete the picture, we compute the quadratic symbols for Q∞ = ∗ ∗ 2 R and Q2. In the first case, R /(R ) = h−1i, and (−1, −1)2,∞ = −1. For p = 2, ∗ ∗ ∗ ∗ 2 2 ∗ 2 ∗ 2 Q2 = h2i × Z2, and Q2/(Q2) = h2 i × (Z2) , now (Z2) = 1 + 8Z2, so we see ∗ ∗ 2 ∼ Q2/(Q2) = h2i × h−1i × h5i. We summarize the values of (x, y)2,2 in the following table: x \ y 2 −1 5 2 1 1 −1 −1 1 −1 1 5 −1 1 1 Proof of the product formula. Consider the following diagram: 0 Y ∗ M pn Kp / Gal(Kp( β)/Kp) p≤∞ p≤∞ √ J / Gal(K( n β)/K) The left vertical map is the definition of the id`eles. The top map is α 7→ ((α, β)n,p)p, which one sees is trivial for almost all p. The horizontal bottom arrow is the Artin Q map. The right vertical map is (σp)p 7→ p σp. ∗ The statement is then that for α ∈ K ⊂ J,√ the global Artin map (the ∗ n bottom arrow) factors as J → J/K = CK → Gal(K( β)/K); this is a nontrivial statement from class field theory.