By Nancy Childress (auth.)

Class box conception, the examine of abelian extensions of algebraic quantity fields, is without doubt one of the biggest branches of algebraic quantity thought. It brings jointly the quadratic and better reciprocity legislation of Gauss, Legendre, and others, and greatly generalizes them. a few of its results (e.g., the Chebotarev density theorem) practice even to nonabelian extensions.

This ebook is an available creation to classification box thought. It takes a conventional procedure in that it offers the worldwide fabric first, utilizing a number of the unique suggestions of facts, yet in a manner that's purifier and extra streamlined than such a lot different books in this topic.

It might be used for a graduate direction on algebraic quantity thought, in addition to for college students who're attracted to self-study. The e-book has been class-tested, and the writer has integrated workouts through the text.

Professor Nancy Childress is a member of the math college at Arizona kingdom University.

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**Example text**

36 2 Dirichlet’s Theorem on Primes in Arithmetic Progressions Proof. Let (m, a) = 1 and consider all Dirichlet characters χ that are defined modulo m. Then χ(a)−1 log L(s, χ) = χ∈(Z/mZ ) χ(a)−1 χ × p 1 ps = p 1 ps = p χ( p) + β(s, χ) ps χ(a)−1 χ( p) + χ χ(a)−1 β(s, χ) χ χ( pa −1 ) + χ χ(a)−1 β(s, χ). χ But, by the orthogonality relations, χ( pa −1 ) = χ ϕ(m) 0 p ≡ a (mod m) otherwise, so ⎛ χ(a)−1 log L(s, χ) = ϕ(m) χ p−s p≡a (mod m) ⎞ something + ⎝ abs. conv. ⎠ . for Re (s) > 12 (∗) Now let s → 1.

We conclude that Y1 = Gal (K /F)⊥ by the bijective correspondence, and X Gal (K /Q) = Gal (K /F) Y1 = Gal (K /F)⊥ ∼ = Gal (K /F), which is isormorphic to the decomposition group for p in K /Q. As mentioned earlier, Dirichlet characters will play an important role in the next section, where we give a proof of Dirichlet’s Theorem on Primes in Arithmetic Progressions. But results such as the one above also may be used to study certain finite unramified abelian extensions of a number field K (in the case when K is abelian over Q).

B. What does your answer to part a tell you about the quadratic subfield(s) of Q(ζ p ), where p is an odd prime? Does a quadratic subfield always exist? Is it unique? When is it real? c. Let p be an odd prime. How many quadratic Dirichlet characters modulo 4 p are there? How many of them are primitive? What does this tell you about the quadratic subfield(s) of Q(ζ4 p ), where p is an odd prime? d. Answer similar questions about the quadratic subfield(s) of Q(ζ8 ). √ e. For any odd prime p, show Q( p) ⊆ Q(ζm ) for m = p or 4 p.