# Faculty of Mathematics

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.