Class Field Theory (Universitext) by Nancy Childress

By Nancy Childress

Classification box thought, the research of abelian extensions of algebraic quantity fields, is likely one of the greatest branches of algebraic quantity idea. It brings jointly the quadratic and better reciprocity legislation of Gauss, Legendre, and others, and enormously generalizes them. a few of its outcomes (e.g., the Chebotarev density theorem) follow even to nonabelian extensions.

This booklet is an obtainable advent to category box thought. It takes a standard technique in that it provides the worldwide fabric first, utilizing the various unique suggestions of facts, yet in a way that's purifier and extra streamlined than so much different books in this topic.

It will be used for a graduate path on algebraic quantity thought, in addition to for college kids who're drawn to self-study. The booklet has been class-tested, and the writer has integrated routines during the textual content.

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Write S ≈ T if S ≺ T ≺ S. 21. Let F be a number field. a. Compute δ F (S), where S = {primes p of O F : f (p/p ∩ Z) = 1}. b. Let S and T be arbitrary sets of primes in O F . Prove or disprove: S ≈ T if and only if S and T differ by finitely many elements. 3. Let E and K be number fields, each of which is Galois over Q. Then S K ≺ S E if and only if E ⊆ K . 44 2 Dirichlet’s Theorem on Primes in Arithmetic Progressions Proof. “⇐=” is clear. For “=⇒” suppose S K ≺ S E . Note that S K E = S E ∩ S K .

Show that if any two of δ(S), δ(T), δ(S ∪ T) are finite then so is the third, and that in this case δ(S ∪ T) = δ(S) + δ(T). 1. Let K /Q be Galois, and let S K = { p ∈ Z : p splits completely in K /Q}. Then δ(S K ) = 1 [K : Q] . Proof. Let ζ K (s) = p (1 − Np−s )−1 for Re (s) > 1, be the Dedekind zeta function for K . Consider s ∈ R, s > 1. We have log(1 − Np−s ) log ζ K (s) = − p ∞ 1 −ns Np . n = p n=1 Now log ζ K (s) = log((s − 1)ζ K (s)) + log(1/(s − 1)), so log ζ K (s) ∼ log(1/(s − 1)) ∞ ∼ p n=1 1 −ns Np n ∞ Np−s + ∼ p ∼ p Np n=2 1 Np−ns n −s p since p ∞ 1 −ns n=2 n Np log ζ K (s) ∼ log 1 s−1 is bounded as s → 1+ .

Example. 10. In K = Q(ζ12 ), the subfield L = Q(i ) has associated characters X L = {χ mod 12 : χ(σ ) = 1 for all σ ∈ Gal (K /L)}. Now Gal (K /L) = {1, σ } where σ fixes i = ζ123 , hence σ : ζ12 → ζ125 . We find that χ(σ ) = 1 if and only if χ(5) = 1. Thus X L = {1, ϑ}, where ϑ(1) = 1, ϑ(5) = 1, ϑ(7) = −1, ϑ(11) = −1. Now ϑ has conductor 4, so ϑ( p) = 0 if and only if p = 2. This confirms that 2 is the only ramified prime in Q(i )/Q. 7. In this exercise we study quadratic Dirichlet characters and their associated fields.

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