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# modules

Module title: Groups, Rings and Fields MTH2010 Prof Mohamed Saidi 15

In this module, you will explore some of the key techniques of modern algebra, including groups, rings, and fields. These topics have their roots in the desire to solve certain equations that arise from arithmetic and geometry.  The most familiar example of a ring is the set of all integers Z=...,-3,-2,-1,0,1,2,3... equipped with the usual operations of addition and multiplication. The familiar properties of these operations serve as a model for the axioms for rings. We can consider whether certain equations have solutions in rings such as the integers. For example, Fermat's Last Theorem famously asserts that if n is a fixed integer that is at least 3, then the equation x^n + y^n = z^n has no solutions for which x, y and z are non-zero integers. Though this problem is easy to state, its solution is extremely difficult: it was first stated in 1637 but the first complete and correct proof was given in 1994. Ring theory is essential for the fourth year module MTHM028 Algebraic Number Theory, which in turn lays the foundations for solving problems such as Fermat's Last Theorem.

Fields are special types of ring in which every non-zero element has a multiplicative inverse. Examples include the rational numbers Q, the real numbers R and the complex numbers C. Group theory was introduced in the first year and will be developed further in this module. Not only does group theory underpin ring theory, but is also interesting and useful in its own right. For example, we shall see that group actions can be used to solve certain counting problems. The material in this module is essential for the study of many of our pure mathematics modules at levels 3 and M, including MTH3004 Number Theory, MTH3026 Cryptography, MTH3038 Galois Theory, MTHM010 Representation Theory of Finite Groups, MTHM028 Algebraic Number Theory and MTHM029 Algebraic Curves. Prerequisite module: MTH1001 (or equivalent).