Math 361, Spring 2013, Assignment 4

I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

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

1. Maximal ideal.
2. Prime ideal.
3. Prime field.
4. Principal ideal domain.
5. Field extension.
6. Algebraic element (of a given field extension).
7. Transcendental element (of a given field extension).
8. Minimal polynomial (of an algebraic element).

Carefully state each of the following theorems:

1. Theorem on quotients by maximal ideals (Theorem 27.9).
2. Theorem on quotients by prime ideals (Theorem 27.15).
3. Theorem relating maximal ideals to prime ideals (Corollary 27.16).
4. Classification of ideals in in \(F[x]\) (Theorem 27.24).
5. Kronecker's Theorem (Theorem 29.3).

Solve the following problems:

• Section 27, problems 3, 5, 15, 19.
• Section 29, problems 3, 5, 7, 9, 17.