It will be important to understand the subfields of a given field; this is a key step in figuring out whether a field is an extension by radicals or not. We need some “machinery” before we can develop this understanding.

Recall:

Definition 1Aringis a set together with two binary operations on such that:

- is commutative.
- There is an additive identity
- Any has an additive inverse
- is associative.
- is associative.
- distributes over both on the right and on the left.