Name: Note for Algebra and Number Theory ANT By Ankita Singh
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Author: Abhishek Apoorv

Notations

Introduction

Basic definitions, Ideals in products of rings, Noetherian rings,  Noetherian modules,Local rings, Rings of fractions ,  The Chinese remainder theorem, Review of tensor products

Preliminaries From Commutative Algebra

First proof that the integral elements form a ring, Dedekind’s proof that the integral elements form a ring Integral elements , Review of bases of A-modules, Review of norms and traces , Review of bilinear forms, Discriminants, Rings of integers are finitely generated, Finding the ring of integers,Algorithms for finding the ring of integers

Rings Of Integers

Discrete valuation rings, Dedekind domains, Unique factorization of ideals, The ideal class group, Discrete valuations, Integral closures of Dedekind domains, Modules over Dedekind domains , Factorization in extensions, The primes that ramify

Dedekind Domains Factorization

Norms of ideals , Statement of the main theorem and its consequences,  Lattices, Some calculus, Finiteness of the class number, Binary quadratic forms

The Finiteness Of The Class Number

Statement of the theorem, Proof that UK is finitely generated, Computation of the rank, S -units, discriminant Finding,  Finding a system of fundamental units , Regulators

The Unit Theorem

The basic results, Class numbers of cyclotomic fields, Units in cyclotomic fields, The first case of Fermat’s last theorem for regular primes

Cyclotomic Extensions Fermats Last Theorem

Nonarchimedean absolute values, Equivalent absolute values, Properties of discrete valuations, Complete list of absolute values for the rational numbers The primes of a number field, The weak approximation theorem, Completions in the nonarchimedean case, Extensions of nonarchimedean absolute values, Newton’s polygon, Locally compact fields

Note for Algebra and Number Theory - ANT By Ankita Singh