Math 361, Spring 2020, Assignment 11

Read:

1. Section 29.

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

1. Field extension.
2. Extension field.
3. Base field (or ground field).
4. Trivial extension.
5. Prime subfield (of a field of characteristic zero).
6. Prime subfield (of a field of characteristic $p$).
7. Algebraic element (of a field extension).
8. Transcendental element.
9. $\mathrm{ann}_{F[x]}(e)$ (the annihilator of $e$ in $F[x]$).
10. $\mathrm{irr}(e,F)$ (the minimal polynomial of $e$ over $F$).

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

1. Kronecker's Theorem.
2. Theorem concerning the irreducibility of $\mathrm{irr}(e,F)$.
3. Theorem relating any irreducible element of $F[x]$ that annihilates $e$ to $\mathrm{irr}(e,F)$.

Solve the following problems:

1. Section 29, problems 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 14.
2. Let $p$ be any (positive) prime integer, and let $n$ be any positive integer at all. Compute $\mathbb{irr}(\sqrt[n]{p},\mathbb{Q})$. Prove your answer. (Hint: at a certain point you will need Eisenstein's criterion to complete the proof.)
3. Compute $\mathrm{irr}(\sqrt[3]{8},\mathbb{Q})$. Does your answer contradict the previous result? At what point would your proof of the previous result break down in this case?

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