Math 361, Spring 2019, Assignment 10

Read:

1. Section 29.

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

1. Algebraic element (of an extension $F\rightarrow E$).
2. Transcendental element (of an extension $F\rightarrow E$).
3. $\mathrm{ann}_{F[x]}(\alpha)$ (the annihilator of $\alpha\in E$).
4. $\mathrm{irr}(\alpha, F)$ (the minimal polynomial of $\alpha$ over $F$).
5. Algebraic extension.
6. Subextension.
7. Subextension generated by a subset.
8. Finitely generated extension.
9. Simple extension.

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

1. Theorem concerning irreducibility of $\mathrm{irr}(\alpha, F)$.
2. Theorem concerning which monic irreducible polynomials can annihilate $\alpha$.

Solve the following problems:

1. Section 29, problems 1, 3, 5, 7, 9, 11, 13, 15, and 17.

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