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

- Basic Algebra: for Algebras, Algebraic Geometry, Algebraic Number Theory
- Representation Theory: for Lie Algebras and their Representations, Representation Theory

## Basic Algebra

You will need this for the following Part III courses:

- Algebras
- Algebraic Geometry
- Algebraic Number Theory

Relevant undergraduate courses are:

### Reality check

- A first course on Groups - you should be comfortable in working with abstract groups and examples. See e.g. Ia Groups. You can check if you are at the required level by going through the example sheets of that course. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.
- The definition of a vector space. See e.g. Ib Linear Algebra. You can check if you are at the required level by going through the example sheets of that course. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

### Prerequisites

You need to be comfortable working with rings and modules, particularly rings of polynomials. For example, you should know:

- definitions and examples of: rings, subrings, ring homomorphisms, ideals, quotient rings, prime ideals, maximal ideals;
- factoring in rings: definitions and properties of zero-divisors, units, prime, and irreducible elements;
- Zorn's lemma and constructing prime ideals;
- examples and counterexamples relating fields, Euclidean domains, principal ideal domains, unique factorization domains, and integral domains; the statement of Gauss' lemma;
- definitions and examples of: modules, submodules, module maps, quotient modules, direct sums of modules, free modules;
- how to work with polynomial rings.

You can use the following exercises to check your present level and to guide your study. For the Example Sheet questions, we recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

- Exercises in the catch-up workshop notes below.
- GRM 2015-16 Example Sheet 2

### Useful books and resources

- Notes from catch-up workshop 2016, provided by Stacey Law, thank you!
- P.J. Cameron
*Introduction to Algebra*, OUP. - B. Hartley, T.O. Hawkes
*Rings, Modules and Linear Algebra: a further course in algebra*, Chapman and Hall, 1970.

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## Representation Theory

You will need this for the following Part III courses:

- Representation Theory
- (Lie Algebras and their Representations)

Relevant undergraduate courses are:

### Reality check

Standard undergraduate level group theory (especially normal subgroups, conjugacy classes in the symmetric group). See e.g. Ia Groups. You can check if you are at the required level by going through the example sheets of that course. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

### Prerequisites

Basics of the representation theory of finite groups over the complex numbers; representations of the symmetric groups. For example:

- Basic definitions and examples of representations, subrepresentations, irreducible representations, the group algebra and permutation representations.
- Schur's Lemma and some consequences. Maschke's theorem.
- Character Theory. More specifically, orthogonality and completeness of characters. The character table of a group.
- Lifting and Induction. Induced characters and Frobenius reciprocity.

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

- Exercises from the Representation Theory workshop 2015, provided by Nicolas Dupré (thank you!)

### Useful books and resources

- Lecture notes for the Part II course (as given in 2011), courtesy of Gareth Taylor.
- Simon Wadsley's online lecture notes for the Part II course (as given in 2012).
- Workshop notes the Representation Theory workshop 2015, provided by Nicolas Dupré (thank you!).
- G. D. James, M. W. Liebeck,
*Representations and characters of groups*, Cambridge University Press, 1993. - Lecture notes for the 2013 Part III course given by Stuart Martin, courtesy of Gareth Taylor (so you can see what level you might be expected to start at).