Part III (MMath/MASt)

The following number theory Part III courses will be offered during the 2021-22 academic year (pre-requisites in brackets):

• Five ways to think about primes (Complex Analysis, Basic Linear Analysis, Elementary Number Theory)
• Local Fields (Galois Theory; Groups, Rings & Modules; Metric and Topological Spaces)
• Elliptic Curves (Basic Algebraic Geometry; Galois Theory; Number Fields)
• Modular Forms and L-functions (Complex Analysis)
• For Local Fields, exposure to Number Fields is desirable but not essential.

## Basic Algebraic Geometry

The Algebraic Geometry course webpage is here.

For the Elliptic Curves course, it will help to have some prior exposure to the basic notions of Algebraic Curves (as for example is covered in the second half of the undergraduate course). For alternative sources see the Elliptic Curves course description.

## 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 (not required for Five ways to think about primes),
• Branch cuts.

## Metric and Topological Spaces

For Local Fields, a basic understanding of metric spaces is essential. The relevant course webpages are here and here.

Reality Check:

• Open and closed subsets, continuity, homeomorphisms.
• Compactness; every closed subset of a compact space is compact.
• Metric spaces; topology defined by a metric.
• A metric space is complete if every Cauchy sequence converges.

Prerequisites

• Topological spaces
• Metric spaces
• Completeness and compactness.

## Basic Linear Analysis

The course webpage is here.

Reality Check:

• Definition of a normed space.
• Definition of an inner product.
• Weierstrass approximation theorem.

For 'Five ways to think about primes', only finite dimensional normed spaces will appear.

The following topic with be proved ab initio in the course, but prior exposure is desirable:

• Hyperplane separation theorem

## Elementary Number Theory

The Part II Number Theory course webpage is here.

Reality check:

• Greatest common divisors and least common multiples.
• Fundamental Theorem of Arithmetic.

In 'Five ways to think about primes', the following topics will be introduced ab initio, but it will be useful to have met them (at least briefly) before:

• Dirichlet convolution
• Möbius function
• von Mangoldt function
• Multiplicative function (definition)