## 403/503 – Eigenvectors for operators on real odd dimensional spaces

February 28, 2010

The goal of this note is to give a proof of the following result:

Theorem 1 Let ${V}$ be an odd dimensional vector space over ${{\mathbb R},}$ and let ${T:V\rightarrow V}$ be linear. Then ${T}$ admits an eigenvector.

The proof that follows is in the spirit of Axler’s textbook, so it avoids the use of determinants. However, I feel it is easier than the argument in the book, and it has the additional advantage of not depending on the fundamental theorem of algebra. In fact, the motivation for finding this argument was to avoid the use of the fundamental theorem.

The proof we present of Theorem 1 can be seen as an elaboration of the argument in the case when ${{\rm dim}(V)=3,}$ that we discussed in lecture. It was found by David Milovich in Facebook.

## 305 -6. Rings, ideals, homomorphisms

March 13, 2009

It will be important to understand the subfields of a given field; this is a key step in figuring out whether a field ${{\mathbb Q}^{p(x)}}$ is an extension by radicals or not. We need some “machinery” before we can develop this understanding.

Recall:

Definition 1 A ring is a set ${R}$ together with two binary operations ${+,\times}$ on ${R}$ such that:

1. ${+}$ is commutative.
2. There is an additive identity ${0.}$
3. Any ${a}$ has an additive inverse ${-a.}$
4. ${+}$ is associative.
5. ${\times}$ is associative.
6. ${\times}$ distributes over ${+,}$ both on the right and on the left.