Math 361, Spring 2019, Assignment 9

Read:

1. Section 27.

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

1. Maximal ideal.
2. Prime ideal.
3. Field extension.
4. Base field.
5. Extension field.
6. Injection (occuring in a field extension).

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

1. Theorem relating maximal ideals to fields.
2. Theorem relating prime ideals to integral domains.
3. Theorem relating maximal ideals to prime ideals.
4. Classification of ideals of $F[x]$.
5. Theorem characterizing when $F[x]/\left\langle m\right\rangle$ is a field.

Solve the following problems:

1. Section 27, problems 1, 3, 5, 7, 9, 15, 16, 17, 18, and 19.

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