Please take this page in conjunction with the Part III Guide to Courses Number Theory section.

- Basic Algebra: for Algebraic Number Theory
- Galois Theory: for Algebraic Number Theory, Modular Forms and L-functions, (Commutative Algebra)
- Number Fields: for Algebraic Number Theory, (Commutative Algebra)

## Basic Algebra

This section is on the Algebra page.

## Galois Theory

You will need this for the following Part III courses:

- Algebraic Number Theory
- Modular Forms and L-functions
- (Commutative Algebra)

Relevant undergraduate courses are:

### Reality check

For most of the below, see Ib Groups, Rings and Modules.

- Polynomial rings. If
*K*is a field then*K*[*X*] is a Euclidean domain (and so a principal ideal domain). Let*f(X)*∊*K*[*X*] be an irreducible polynomial, then the ideal*<f(x)>*is maximal. - Division Algorithm of polynomials. Let
*K*be a field, and*f(X)*∊*K*[*X*] be a polynomial of degree*n*. Then for any polynomial*g(X)*∊*K*[*X*], there exist*q(X),r(X)*∊*K*[*X*] such that*g(X)=f(X)q(X)+r(X)*where either*r(X)=0*or deg*r(X)*< deg*f(X)*. - Gauss's lemma and Eisenstein criterion about irreducibility of polynomials in ℤ[
*X*] and ℚ*[X]*: For each primitive polynomial*f(X)*∊*[X]*⊂ℚ*[X]*,*f(X)*is irreducible in ℤ[*X*] if and only if*f(X)*is irreducible in ℚ[*X*]. - Eisenstein criterion: Let
*f(X)=a*∊ℤ be a primitive polynomial with integer coefficients. If there exists a prime number_{n}X^{n}+a_{n-1}X^{n-1}+...+a_{1}X+a_{0}, a_{i}*p*such that*p*does not divide*a*,_{n}*p*divides*a*for each_{i}*i*≠*n*, and*p*does not divide^{2}*a*, then_{0}*f*is irreducible. - Any non-trivial field homomorphism is injective.
- Applications of the rank-nullity theorem:
- Any finite integral domain is a field.
- Let
*L*be vector spaces over_{1}, L_{2}*K*such that dimdim_{K}(L_{1})=. If_{K}(L_{2})*s: L*→_{1}*L*is an injective_{2}*K*-linear map, then*s*is an isomorphism.

### Prerequisites

Galois Theory topics such as:

- field extensions,
- tower law,
- algebraic extensions,
- separability and primitive element theorem,
- automorphism of fields,
- Galois extension,
- fundamental theorem of Galois,
- finite fields,
- cyclotomic extensions,
- Kummer theory.

You can use the following exercises to check your present level and to guide your study.

- Excercises within these Galois Theory Workshop notes;
- additional exercises, both provided by Zexiang Chen (thank you!).

### Useful resources

- The workshop notes given above.
- Galois Theory Notes by Dr. T.Yoshida

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## Number Fields

You will need this for the following Part III courses:

- Algebraic Number Theory
- (Modular Forms and L-functions)
- (Commutative Algebra)

Relevant undergraduate courses are:

### Reality check

- Algebraic numbers and algebraic integers
- Number fields and their ring of integers
- Some examples, e.g. the Gaussian integers

You can check whether you are at the right level by trying the first half of this example sheet: Example Sheet 1 from Part II Number Fields (2015/16).

### Prerequisites

- Failure of unique prime factorization (in general number fields)
- Unique factorization into prime ideals
- Class groups (including finiteness)

### Useful resources

- Notes based on Part II Number Theory, written by Zexiang Chen (thank you!) based on the lectures by Dr Fisher, with a few additions. They contain exercises for you to try.
- Notes based on Part II Number Fields, written by Zexiang Chen (thank you!) based on the Cambridge lecture course, with more significant additions. They contain exercises for you to try.

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