Math 361, Spring 2018, Assignment 7

Carefully define the following terms, then give one example and one non-example of each:

1. Prime subfield.

Carefully state the following theorems (you do not need to prove them):

1. Theorem relating prime ideals to integral domains.
2. Theorem relating maximal ideals to prime ideals.
3. Description of the prime ideals of $F[x]$.
4. Theorem concerning the existence of prime subfields.
5. Example of an infinite field of characteristic $p$.

Solve the following problems:

1. What is the prime subfield of $\mathbb{R}$?
2. What is the prime subfield of $\mathbb{Q}$?

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