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Part III (MMath/MASt)

 

The following number theory Part III courses will be offered during the 2019-20 academic year (pre-requisites in brackets):

  • Algebraic Number Theory (Galois Theory; Groups, Rings & Modules)
  • Elliptic Curves (Galois Theory; Number Fields)
  • Modular Forms and L-functions (Complex Analysis)
  • Analytic Number Theory (Complex Analysis)
 

Galois Theory

The course webpage is here.

Reality check

  • 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) ∊ℤ[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 extensions,
  • 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.

 

Groups, Rings & Modules

The course webpage is here.

Reality check

  • The first, second and third isomorphism theorems for each of 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) ∊ℤ[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.

Prerequisites

Groups, Rings & Modules topics such as:

  • Algebraic integers,
  • Noetherian rings and the Hilbert basis theorem,
  • Structure theorem for finitely generated modules over a PID.

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

  • Exercises within these Groups, Rings & Modules Workshop notes provided by Stacey Law (thank you!).
 

Number Fields

The course webpage is here.

Reality check

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

Prerequisites.

Number Fields topics such as:

  • Norm and trace of an algebraic integer,
  • Statement of unique factorization of ideals into prime ideals,
  • Definition of the ideal class group (including finiteness),
  • Statement of Dirichlet’s unit theorem.

Useful resources

  • Lecture notes and example sheets from the 2019 Number Fields course are available here.
 

Complex Analysis

The course webpage is here.

Reality check

  • The definition of a holomorphic function.
  • Taylor expansion and Laurent expansion of a holomorphic function on a (punctured) disc.

Prerequisites.

Complex Analysis topics such as:

  • Contour integration and the Cauchy integral formula,
  • Residue theorem,
  • Maximum modulus principle,
  • Branch cuts.

Useful resources

  • A sheet detailing what you need to know for the Analytic Number Theory lecture course is available here.