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 deg*r(X)*< deg*f(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)=a*∊ℤ be a primitive polynomial with integer coefficients. If there exists a prime number_{n}X^{n}+a_{n-1}X^{n-1}+...+a_{1}X+a_{0}, a_{i}*p*such that*p*does not divide*a*,_{n}*p*divides*a*for each_{i}*i*≠*n*, and*p*does not divide^{2}*a*, then_{0}*f*is irreducible. - Any non-trivial field homomorphism is injective.
- Applications of the rank-nullity theorem:
- Any finite integral domain is a field.
- Let
*L*be vector spaces over_{1}, L_{2}*K*such that dim_{K}*(L*dim_{1})=_{K}*(L*. If_{2})*s: L*→_{1}*L*is an injective_{2}*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.

- Exercises within these Galois Theory
__Workshop notes__; __additional exercises__, both provided by Zexiang Chen (thank you!).

**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 deg*r(X)*< deg*f(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)=a*∊ℤ be a primitive polynomial with integer coefficients. If there exists a prime number_{n}X^{n}+a_{n-1}X^{n-1}+...+a_{1}X+a_{0}, a_{i}*p*such that*p*does not divide*a*,_{n}*p*divides*a*for each_{i}*i*≠*n*, and*p*does not divide^{2}*a*, then_{0}*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.