Algebraic Numbers and Algebraic Functions
This book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.
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abelian group algebraic extension algebraic function field algebraic integer algebraic number field algebraically closed automorphism basis branch points coefficients complete convex subgroup Corollary corresponding Dedekind domain defined denote differential divisor class divisor of poles elliptic function field equation factorization field of fractions field of genus follows fractional ideal function field K/k given global field ground field hence holds homomorphism integral closure integral divisor integral domain isomorphism Lemma Let K/k linear linearly independent maximal ideal multiplication non-archimedean non-special non-trivial non-zero norm obtain ordered group p-adic prime divisors prime ideal principal valuation product formula Proof Let Proposition ramification rational function rational function field real numbers residue class field ring of integers satisfies separable extension separating element subring Theorem 4.1 topology unique valuation ring value group Weierstrass points write zero