# Lecture notes- Galois Theory

To save carrying the original paperwork about, and to give myself a recap on the material, I’ve written up the lecture notes from MA40037:Galois Theory as taught at the University of Bath by Geoff Smith.

The content is broadly as follows: Rings, Integral Domains, Fields of Fractions, Units, Ideals, Homomorphisms, The First Isomorphism Theorem, The Chinese Remainder Theorem, Irreducibles, Field Extensions, Characteristic, Minimal Polynomials and Algebraic Numbers, Galois Theory.

The notes very closely match those I made and hence the lectures given, except the section on the Chinese Remainder Theorem, which was adapted from problem sheets. There have been various minor linguistic tweaks, but few mathematical ones.

It should be noted (to avoid confusion under composition) that the convention of writing function arguments to the left (i.e., (x)f rather than f(x)) is adopted here; and that square brackets are sometimes used for factors in polynomials where these appear in expressions also featuring function or polynomial evaluations (which are denoted by round brackets).

1. Hi,

Sorry, I have not read through the whole document (since I haven’t been ‘formally’ taught all that, so most went whoosh over my head anyway!) But although it may be obvious from the context, do you mean that [x] is something to do with equivalence classes or the integer part? (near the integral rings part).

BTW- I know this might sound a bit cheeky off me, but for a LaTeX newbie, (if it’s not too much of a hassle, and if you don’t mind); I was wondering whether you could send me the tex file of that document. (It’s just that a lecturer told me that a good way to become better at using LaTeX, is to read through other peoples tex documents.) Yours looks rather neat too. ðŸ˜®

Also, would you recommend typing my notes up using LaTeX? Did you find it useful?

2. Yes, those are equivalence classes- I remember being amazed how much they crop up in abstract algebra, considering how neglected they tend to be earlier in the undergrad curriculum. The year I was tutoring the appropriate first-year course at Bath they were relegated to optional reading, which will hurt any would-be algebraists when they attempt the Galois course.

I’ve emailed the LaTeX source to the yahoo account you used to comment, so I hope that works!

3. We were taught equivalence classes in the reasoning module (sets, numbers and functions), but I confess to having forgotten about them. I better go and reread my notes about them!

I have received the email – many thanks for sending it. ðŸ™‚

BTW, I am assuming that Galois Theory is a fourth year course, but do you reckon a third year student could formally study the module?

4. At Bath Galois Theory ran every other year, so half the group were third year students. Including me, and I didn’t drop a mark in the exam, so it’s definitely feasible!