skip to content
 

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)

Back to main page


Basic Algebra

This section is on the Algebra page.

Back to top


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 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.

Useful resources

Back to top; Back to Algebra


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.

Back to top; Back to Algebra