# Lemma 1.1. The field K embeds as a subfield of Q(ζ D ).

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2 2 2. Frobenius elements Now we turn our attention to the computation of χ K (p) for positive primes p D. The computation of χ K ( 1) rested on identifying the Galois automorphism 1 mod D on Q(ζ D ) with complex conjugation, and the fact that this restricts to complex conjugation on quadratic subfields. We require an analogous interpretation of p mod D as a Galois automorphism of Q(ζ D ) in a manner that is well-behaved with restriction to quadratic subfields. The interpretation will rest on Frobenius elements. Since p D, pz is unramified in Z[ζ D ]. Thus, for any p over p in Z[ζ D ] we get (by Exercise 4(v) in Homework 10) a canonical Frobenius element φ p pz in Gal(Q(ζ D )/Q) that generates D(p pz) = Gal(Q(ζ D ) p /Q p ) and is uniquely characterized in this decomposition group via the condition that on the residue field κ(p) it induces the automorphism x x #κ(pz) = x p. Recall the following general behavior of decomposition groups and Frobenius elements with respect to conjugation: Lemma 2.1. Let K /K be a Galois extension of a global field K and let v be a non-archimedean place on K with v a place over v on K. For g Gal(K /K), let g(v ) be the place on K over v given by x g(v ) = g 1 (x ) v (so in the case that K /K is finite with v arising from a prime ideal p v of the integral closure of the uncompleted discrete valuation ring O K,v, g(v ) arises from the prime ideal g(p v )). We have and the resulting identification D(g(v ) v) = gd(v v)g 1, I(g(v ) v) = gi(v v)g 1, D(g(v ) v)/i(g(v ) v) = gd(v v)g 1 /gi(v v)g 1 carries φ(g(v ) v) to gφ(v v)g 1. In particular, if Gal(K /K) is abelian then the subgroups D(v v) and I(v v) in Gal(K /K) are independent of the choice v over v, and the element φ(v v) D(v v)/i(v v) is independent of v over v. Proof. This is a simple exercise in unwinding definitions, as well as using the unique characterization of the Frobenius element via its effect on residue fields. (In particular, one uses that if q = p a with a > 0 then the qth-power map is functorial with respect to all maps between commutative F p -algebras.) The most important case of Lemma 2.1 is when v is unramified in K, in which case I(v v) = 1 and hence φ(v v) is an element of Gal(K /K) whose conjugacy class only depends on v. Due to this lemma, in the case of abelian extensions of a global field we usually write D v, I v, and φ v rather than D(v v), I(v v), and φ(v v), and we call these respectively the decomposition group at v, the inertia group at v, and the (relative) Frobenius element at v in Gal(K /K). Let n be a nonzero integer. For any positive prime p n, we let Frob p denote the Frobenius element at pz in the abelian Galois group Gal(Q(ζ n )/Q). This element fixes every prime p over pz and induces the pth-power automorphism on κ(p) = Z[ζ n ]/p because #κ(pz) = p (since p > 0). Lemma 2.2. For any nonzero integer n and any positive prime p n, under the isomorphism Gal(Q(ζ n )/Q) (Z/nZ) the Frobenius element Frob p at the prime pz goes over to p mod n. Observe that p mod n p mod n for n > 2, so the description of the Frobenius element as a specific residue class modulo n is sensitive to the distinction between the two generators ±p of pz. Proof. By the definition of the isomorphism to (Z/nZ), the automorphism σ p giving rise to the residue class p mod n acts on Z[ζ n ] via ζ n ζn. p We pick a prime p over p and we need to show that σ p (p) = p and that the automorphism induced by σ p on the finite field κ(p) = Z[ζ n ]/p is the pth-power map. The endomorphism induced by σ p on the F p -algebra Z[ζ n ]/(p) = F p [T ]/(Φ n (T )) sends T to T p, and so it must be the pth-power map. This map fixes all idempotents, and so the bijection between prime factors of (p) and primitive idempotents of Z[ζ n ]/(p) implies that σ p fixes all primes p over pz. Moreover, on the quotient κ(p) of Z[ζ n ]/(p) the automorphism induced by σ p must clearly be the pth-power map, so σ p = φ p pz as desired.

3 3 To exploit the fact that the isomorphism Gal(Q(ζ D )/Q) (Z/DZ) carries Frob p to p mod D for positive primes p D, we need to see how Frobenius elements behave with respect to quotients of Galois groups. Lemma 2.3. Let K /K /K be a tower of finite extensions of global fields, with K and K each Galois over K. If v on K is a non-archimedean place over places v on K and v on K, then the quotient map Gal(K /K) Gal(K /K) carries D(v v) onto D(v v) and carries I(v v) into I(v v), with the induced map D(v v)/i(v v) D(v v)/i(v v) carrying φ(v v) to φ(v v). In particular, if v is unramified in K then Gal(K /K) Gal(K /K) carries φ(v v) to φ(v v). Note that we do not claim that I(v v) maps onto I(v v); this is related to the fact that κ(v ) may be strictly larger than κ(v ). The final part of this lemma is sometimes referred to as the functoriality of the Frobenius element with respect to passage to quotients. Proof. There is an induced tower K v /K v /K v of completions, and these are Galois because K v = K K v and K v = K K v (why?). Moreover, the inclusions of decomposition groups into the global Galois groups are identified with the natural maps of Galois groups and it is easy to check that the diagram Gal(K v /K v) Gal(K /K), Gal(K v /K v) Gal(K /K), Gal(K v /K v) Gal(K /K) Gal(K v /K v) Gal(K /K) commutes. The left side is surjective by Galois theory, and so D(v v) D(v v) is surjective. The natural surjective map Gal(K v /K v) Gal(K v /K v) of Galois groups of local fields is compatible with the induced map Aut(κ(v )/κ(v)) Aut(κ(v )/κ(v)) and so it carries I(v v ) into I(v v) and identifies the induced map of quotients with the natural map of Galois groups D(v v)/i(v v) D(v v)/i(v v) Gal(κ(v )/κ(v)) Gal(κ(v )/κ(v)). Hence, the desired behavior with respect to Frobenius elements is a consequence of the obvious general fact that if k /k /k is a tower of finite fields with q = #k then the surjective map Gal(k /k) Gal(k /k) carries the qth-power map to the qth-power map. The preceding lemma implies that the natural quotient map Gal(Q(ζ D )/Q) Gal(K/Q) carries Frob p to the Frobenius element φ K,p for the prime pz that is unramified in K. In general, for any finite Galois extension F /F of global fields and any non-archimedean place v of F that is unramified over its restriction v in F, the order of φ(v v) in Gal(F /F ) is the residual degree f(v v) because φ(v v) is a generator of the cyclic group Gal(κ(v )/κ(v)) of order f(v v). In particular, φ(v v) is trivial if and only if f(v v) = 1. As a special case, if [F : F ] = 2 then an non-archimedean place v of F that is unramified in F is split (resp. inert) in F if and only if φ v = 1 (resp. φ v 1). Thus, for a positive prime p D we conclude that the Frobenius element φ K,p Gal(K/Q) is trivial (resp. non-trivial) if and only if pz is split (resp. inert) in O K. In view of the definition of χ K : (Z/DZ) ±1 via Galois groups, we have proved: Theorem 2.4. For a positive prime p D, χ K (p) = 1 if and only if pz is split in O K, and χ K (p) = 1 if and only if pz is inert in O K.

4 4 3. Jacobi symbols By Homework 3, Exercise 3(ii), if p is odd then pz is split in O K if and only if (D p) = 1 and pz is inert in O K if and only if (D p) = 1. By the same exercise, if p = 2 (so D is odd, as p D, so D 1 mod 4) then 2Z is split in O K if and only if D 1 mod 8 and 2Z is inert in O K if and only if D 5 mod 8. Thus, by Theorem 2.4 we obtain: Corollary 3.1. For a positive odd prime p D, χ K (p mod D) = (D p). If D is odd then χ K (2 mod D) = ( 1) (D2 1)/8. Our earlier result that χ K ( 1 mod D) expresses the action of complex conjugation on K is analogous to Corollary 3.1 in the sense that complex conjugation (relative to an embedding into C) is generally considered to be the Frobenius element at a real place (since Gal(C/R) is generated by complex conjugation). Definition 3.2. Let N be a nonzero integer. The Jacobi symbol (N ) is the unique ±1 -valued totally multiplicative function on the set of nonzero integers relatively prime to D such that (N 1) = sign(n), (N p) is the Legendre symbol for positive odd primes p not dividing N, and (N 2) = ( 1) (N 2 1)/8 if N is odd. By definition, clearly (NM n) = (N n)(m n) for nonzero integers n, N, M with gcd(n, NM) = 1. (The only part requiring a check is the case n = 2.) Our preceding work shows that if D is the discriminant of a quadratic field then (D n) = χ K (n mod D) for nonzero integers n relatively prime to D because both sides are totally multiplicative in n and they coincide for n = 1 and for n = p a positive prime not dividing D. This yields a conceptual proof of a non-obvious fact that is often proved in elementary texts by tedious application of quadratic reciprocity: Theorem 3.3. Let N be a nonzero integer and write N = ν 2 N with squarefree N and ν Z +. The Jacobi symbol (N n) only depends on n mod N if N 1 mod 4 and on n mod 4N otherwise. In particular, (N ) is a well-defined quadratic character on (Z/N Z) if N 1 mod 4 and on (Z/4N Z) otherwise. Proof. If N 1 mod 4 then clearly N 1 mod 4. By multiplicativity in N, we have (N n) = (N n) for nonzero n relatively prime to N, so we conclude that it suffices to replace N with N. Hence, we may assume that N is squarefree. Similarly, using the the isomorphism (Z/NZ) (Z/N 1 Z) (Z/N 2 Z) and the equalities (N n) = (N 1 n)(n 2 n) and N = N 1N 2 if N = N 1 N 2 with gcd(n 1, N 2 ) = 1, we may assume that N is prime or N = ±1. The cases N = ±1 are trivial, so it remains to handle exactly one of the cases N = p or N = p for each positive prime p. The case N = 2 is clear by inspection, so it suffices to treat the case N = ( 1 p)p 1 mod 4 for an odd prime p. This case follows from the relationship with χ K for the quadratic field K = Q( N) with discriminant N. We may now summarize our conclusion by means of the commutativity of the diagram: (1) Gal(Q(ζ D )/Q) (Z/DZ) Gal(K/Q) χ D ±1 where χ D = (D ) is a quadratic character on (Z/DZ). Let us conclude with an interesting refinement on the embeddability of K into Q(ζ D ): Theorem 3.4. The cyclotomic field Q(ζ D ) is the smallest one that contains K, in the sense that a cyclotomic field containing K must contain Q(ζ D ). This theorem admits a very simple conceptual proof via local ramification considerations once the machinery of class field theory is available. Proof. Since the intersection Q(ζ n ) Q(ζ m ) inside of an algebraic closure of Q is equal to Q(ζ (n,m) ) (with n, m Z nonzero), it suffices to prove that K is not contained in any proper cyclotomic subfields of Q(ζ D ).

5 5 Recall that Q(ζ n ) = Q(ζ m ) inside of an algebraic closure of Q if and only if either n = m, n = 2 m with odd m, or m = 2 n with odd n. Since D is either odd or a multiple of 4, it follows that a proper cyclotomic subfield of Q(ζ D ) is precisely a cyclotomic field of the form Q(ζ n ) with n a proper (possibly negative) divisor of D. It is therefore necessary and sufficient to show that the quadratic character (D ) on (Z/DZ) does not factor through the projection (Z/DZ) (Z/δZ) for a proper (possibly negative) divisor δ of D. This is now a purely group-theoretic problem. We write D = δδ with δ > 1, and by shifting prime factors into δ we may assume δ is prime. First assume δ is odd, so gcd(δ, δ ) = 1. We may suppose δ = ( 1 p)p for an odd prime p, so δ 1 mod 4. Since (D ) = (δ )(δ ) as functions on the set of integers relatively prime to D, and (δ n) only depends on n mod δ, we conclude that for nonzero n relatively prime to D the function (δ n) only depends on n mod δ. However, since δ is an odd prime we know that (δ n) also only depends on n mod δ. In other words, if (D ) factors through (Z/δZ) then the homomorphism (δ ) : (Z/DZ) ±1 factors through both projections (Z/DZ) (Z/δZ), (Z/DZ) (Z/δ Z), and from this we seek a contradiction. Consideration of primary components of Z/DZ shows that the kernels of these two projections generate (Z/DZ) because gcd(δ, δ ) = 1, and hence (δ n) = 1 for all n relatively prime to D. Since the map (Z/DZ) (Z/δ Z) is surjective, it follows that (δ n) = 1 for all n relatively prime to δ. Since δ = ( 1 p)p 1 mod 4 for an odd prime p, Jacobi reciprocity gives (δ n) = (n δ ) for any odd positive integer n relatively prime to δ. We can find such n representing any nonzero residue class modulo δ, and so in particular by taking a non-square residue class we find such n for which (δ n) = 1. This gives a contradiction. Now it remains to consider the case when δ = ±2, so in particular D = 4d for a squarefree integer d 2, 3 mod 4. We have to deduce a contradiction if (D n) only depends on n mod 2d. By factoring D into a product of even and odd parts, a simple argument as above with the Chinese remainder theorem implies that if d is odd then ( 4 n) only depends on n mod 2 for odd n (that is, ( 4 n) = 1 for all odd n) and that if d is even then (8 n) = (2 n) only depends on n mod 4 for odd n. Since ( 4 1) = 1 and (2 5) = 1, we get a contradiction in both cases. 4. Application to zeta-functions The Riemann zeta function is ζ(s) = n 1 n s for s C with Re(s) > 1, which is absolutely and uniformly convergent in each closed half-plane Re(s) 1 + ε with ε > 0. There is also the well-known convergent Euler product ζ(s) = p (1 p s ) 1 that is also absolutely and uniformly convergent in each closed half-plane Re(s) 1 + ε with ε > 0. On page 283 of the book Algebraic Number Theory by Fröhlich and Taylor, this definition is generalized to any number field K: ζ K (s) = a 0 N(a) s = p (1 N(p) s ) 1 where the sum is taken over all nonzero ideals of O K and the product is taken over all nonzero prime ideals of O K. It is also proved there that these sums and products enjoy the same convergence properties as in the classical case K = Q (ultimately by reduction to the classical case). If K is a quadratic field with discriminant D then for Re(s) > 1 we get the identity ζ K (s) = ζ(s) ( 1 χ ) 1 D(p mod D) p s p D from comparing factors over p for all positive rational primes p, since χ D (p mod D) = 1 for pz split in O K and χ D (p mod D) = 1 for pz inert in O K. We emphasize that p is always understood to denote a positive prime. We can express this factorization of ζ K in terms that are intrinsic to Gal(K/Q) as follows. We let ψ : Gal(K/Q) C be the unique non-trivial character, so by the commutative diagram (1) ψ corresponds

6 6 to χ D via the composite surjection (Z/DZ) Gal(Q(ζ D )/Q) Gal(K/Q). Thus, if we define L(s, ψ) = ( 1 ψ(frob ) 1 K,p) p s p D for Re(s) > 1, with Frob K,p Gal(K/Q) denoting the Frobenius element at p, then for Re(s) > 1. ζ K (s) = ζ Q (s)l(s, ψ)

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