• 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)

Back to list of subject areas

## 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)

### 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 degr(X) < degf(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)=anXn+an-1Xn-1+...+a1X+a0, ai∊ℤ be a primitive polynomial with integer coefficients. If there exists a prime number p such that p does not divide an, p divides ai for each in, and p2 does not divide a0, then f is irreducible.
• Any non-trivial field homomorphism is injective.
• Applications of the rank-nullity theorem:
• Any finite integral domain is a field.
• Let L1, L2 be vector spaces over K such that dimK(L1)=dimK(L2). If s: L1 L2 is an injective 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.

## Number Fields

You will need this for the following Part III courses:

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

### 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.