Vector Spaces

We give definitions for span, basis and dimension, and then prove that all vector spaces have bases, and that their dimension is well-defined. Then we use that to define the degree of a field extension and prove the tower law for field extensions. After that, we define basic properties of polynomials.

In the previous entry, we introduced Rings and Fields. In this entry we continue our investigation. We will investigate structures that can be built on top of them. This entry will mostly involve grinding through fundamental proofs that are necessary for more interesting results which will come later.