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

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Basic Algebra

This section is on the Algebra page.

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Galois Theory

You will need this for the following Part III courses:

  • Algebraic Number Theory
  • Elliptic Curves
  • Modular Forms
  • (Algebras)

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

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Number Fields

You will need this for the following Part III courses:

  • Algebraic Number Theory
  • (Elliptic Curves)
  • (Modular Forms)

Relevant undergraduate courses are:

Reality check

coming soonish

Prerequisites

coming soonish

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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