Abstraction

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The Dirichlet L-functions are an extremely important tool in studying primes in arithmetic progressions – their zeros “control” distribution of primes in arithmetic progressions in the same manner as zeros of Riemann zeta function control the overall distribution of primes. The first and the most elementary result involving these zeros, which is the key result in most proofs of Dirichlet’s theorem, is that there is never a zero at point $$s=1$$. This post will present a proof of this fact using results from algebraic number theory.

This post is based solely on the content of Marcus’s Number Fields. The prerequisities for it are basic results about ideals in number fields and a minute amount of complex analysis. No background in analytic number theory is necessary.

We shall establish relations between degrees, inertia degrees and ramification indices involved in decomposition and inertia field of a given prime in Galois extension. This is based on Marcus’s Number Fields and online notes by R. Ash. It follows a very similar appoach to, but is not based on, the one which can be found in this blog post by Sander Mack-Crane. For this blog post, understanding of Galois theory and basic facts about number fields is necessary.

Based on arguments in Marcus’s Number Fields and K. Conrad’s expository paper on ideal factorization. I assume familiarity with the concepts related to ideals and fractional ideals in a commutative ring.

Recall that an integral domain $$R$$ with the field of fractions $$K$$ is called a Dedekind domain if the following conditions hold:

• $$R$$ is Noetherian, so every nonempty set of ideals has a maximal element, or equivalently, every ideal is finitely generated,
• every prime ideal in $$R$$ is a maximal ideal, and
• $$R$$ is integrally closed in $$K$$, so that every root of a monic polynomial from $$R[x]$$ lying in $$K$$ lies in $$R$$.

Our goal is to prove the following proposition:

Proposition: Assume $$R$$ is a Dedekind domain. For any nonzero (i.e. containing a nonzero element) fractional ideal $$I$$ in $$K$$ there exists a fractional ideal $$J$$ such that $$IJ=R$$.