# Algebraic Number Theory: Home

## Using this Guide to Help with your Research

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## Course Description

A number field K is a finite algebraic extension of the rational numbers Q. Every such extension can be represented as all polynomials in an algebraic number ®: K = Q(®) = (Xm n=0 an®n:an2Q) . Here ® is a root of a polynomial with coefficients in Q. Algebraic number theory involves using techniques from (mostly commutative) algebra and finite group theory to gain a deeper understanding of number fields. The main objects that we study in algebraic number theory are number fields, rings of integers of number fields, unit groups, ideal class groups,norms, traces, discriminants, prime ideals, Hilbert and other class fields and associated reciprocity laws, zeta and L-functions, and algorithms for computing each of the above.

## Algebraic Number Theory

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