## 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.

Last lecture we defined ${{\mathbb F}^{p(x)}}$ where ${{\mathbb F}}$ is a subfield of a field ${{\mathbb K},}$ all the roots of the polynomial ${p(x)}$ are in ${{\mathbb K},}$ and all the coefficients of ${p(x)}$ are in ${{\mathbb F}.}$ Namely, if ${r_1,\dots,r_n}$ are the roots of ${p,}$ then ${{\mathbb F}^{p(x)}={\mathbb F}(r_1,\dots,r_n),}$ the field generated by ${r_1,\dots,r_n}$ over ${{\mathbb F}.}$
The typical examples we will consider are those where ${{\mathbb F}={\mathbb Q},}$ ${{\mathbb K}={\mathbb C},}$ and the coefficients of ${p(x)}$ are rational or in fact, integers.