Week 
Date of Monday 
Topics 
Pages in Siksek 
Example sheet 
Comments 
1 
Oct. 1 
Introduction and overview. Revision: Rings and fields.
Polynomial rings. Homomorphisms, automorphisms, relative
automorphisms, fixed rings. Evaluation homomorphisms. Quotients,
first isomorphism theorem. Principal ideals, polynomial rings
over fields are PIDs. Maximal ideals, fields.

1  6 

There are no support classes this week. Many questions were
asked (partly by the lecturer!) about whether or not rings must
have a unit.

2 
Oct. 8 
Extensions and subfields. Quotients. Fields of fractions.
Generated subfields. Adjoining roots. The subfields of
\(\QQ(\sqrt{2}, \sqrt{3})\). Simple extensions. Adjoining a root
gives the generated subfield (Prop 21). Splitting fields. Fields
of rational functions.

7  12 
One 

3 
Oct. 15 
Existence of splitting fields (uniqueness postponed).
Extensions are vector spaces. Degrees of extensions. Algebraic
and transcendental numbers. Algebraic extensions. Finite
extensions are algebraic. Monic polynomials, minimal polynomials.
Review of Gauss's lemma, Eisenstein's criterion.

12  17 
Two 

4 
Oct. 22 
The structure of simple algebraic extensions (Prop 41, Thm
42). Primitive elements, degrees. The Tower Law. Extended
examples. The algebraic numbers form a field.

17  22 

Questions asked by
students on 20181025. 
5 
Oct. 29 

23  ?? 
Three 

6 
Nov. 5 




7 
Nov. 12 




8 
Nov. 19 




9 
Nov. 26 




10 
Dec. 3 



