## Category Archives: algebra

### Automorphism groups of groups

After last week’s GAGA tutorials, I wrote some notes related to sheet 5.

A quadratic form is essentially the same thing as a symmetric bilinear form, at least over a field of characteristic different from 2. A while ago, I wrote a brief note on quadratic forms dealing with diagonalization and geometry. I hope it’s useful for HT2013, sheet 4, problem 8, in particular.

### Orthogonal bases

I was a bit confused during tutorials today about orthogonal bases, so I wrote up a short note on how to find them. This method is illustrated with problem 5 on sheet 4 for Linear Algebra II.

### Group theory tutorial, TT week 1 (=HT week 9)

I got quite confused with one of the problems in tutorial today. I had a glimmer of the right answer, but then misled myself into saying something quite silly. Anyways, here is a short note on this.

### Symmetries

I have written some notes on symmetries following the Group Theory tutorials this week.

### Collections, Hilary Term, 2012

I’ve marked the scripts for the algebra portion of the Merton collections for Mods Pure Maths, Part A AC1 and Part A AC2.
I am including here brief remarks on the questions. Later, if I have the energy, I will also upload comments on the Banach Spaces paper for Part B students.

### POSTECH notes, algebra.

Here are some notes on

Automorphisms of algebraic number fields

to accompany the lecture on 5 December, 2011.

### Norms of elements and principality

Dear Professor Kim,

Sorry to bombard you with these questions. I have come across a problem on your note ‘some principle ideals’. When we factorize $m(x)=x^3 + x - 1$ modulo 3 we get $(x+1)(x^2-x-1)$ we then associate these factors with the ideals $P_3$ and $P_9$ respectively. When we compute the norm of $x^2-x-1$ we do so by calculating the determinant of the matrix $L_{a^2-a-1}$ and find that the norm is in fact 9, so $P_9$ is a principle ideal. However, we could just have easily used $x^2+2x-1$ or $x^2+2x+2$ and in each case I get a different answer for the determinant. Have I made an error or is there a canonical form of sort that I should be aware of?

First of all, I presume your $x^2-x-1$ etc. are $a^2-a-1$ etc. All the elements you mention do indeed belong to the ideal and can be used as generators *when used together with the element 3*. Indeed they are all all evaluations at $a$ of polynomials that are congruent to $x^2-x-1$ mod $3$. However, this does not mean they are generators on their own. Of course different elements in an ideal $I$ will have different norms in general. However, an element $b\in I$ is a generator *by itself* (making $I$ into a principal ideal), exactly when $|N(b)|=N(I)$. Of course such a $b$ need not exist. I haven’t calculated the norms of the elements you mention, but if their norms come out larger than 9, it merely says they are not generators (again, by themselves), while $a^2-a-1$ is.
A thorny point that comes out of this discussion is that if you had initially presented the ideal as $(3, a^2+2a-1)$, for example, then it might have been harder to see that it is principal.