We prove that Z is universal in the category of rings with identity. Using that, we define the characteristic of a field, and prove that it is well-defined. We then discuss ring ideals a bit more before finishing by showing that all fields with exactly p elements (where p is prime) are isomorphic.
We introduce localization of a ring and the ring of fractions of a ring. We then prove that the ring of polynomials over a field is a Euclidean Domain, and then finish by proving Gauss’ Lemma that a polynomial is irreducible over a ring exactly when it is irreducible over the ring’s field of fractions.
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.
