Module description
Syllabus
Review of the basic theory of rings, polynomials and fields; Eisenstein's Criterion; first properties of finite extensions of fields and their degrees; algebraicity and transcendence; field embeddings and automorphisms; normal extensions; separable extensions; the Galois Correspondence; examples of practical calculation; soluble groups and extensions; (in)solubility of polynomial equations by radical expressions.
Further topics may include finite fields, constructability by straightedge and compass, etc. as time allows.
Prerequisites
You must have taken 6CCM350A Rings and Modules or equivalent
Assessment details
Written examination.
Educational aims & objectives
To develop the theory of finite extensions of fields, culminating in an understanding of the Galois Correspondence. To demonstrate the power of this theory by applying it to the solution of historically significant questions. For instance: for which polynomials can all the roots be written as `radical expressions' (i.e. expressions involving the usual operations of arithmetic together with roots of any degree)? To provide an important tool for further studies in Algebra e.g. Number Theory.
Teaching pattern
Three hours of lectures each week and one hour of tutorial
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists