Math 361, Spring 2021, Assignment 14

Read:

1. Section 29.

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

1. Field extension.
2. Base field (of a field extension $F\rightarrow E$).
3. Extension field (of a field extension $F\rightarrow E$).
4. Injection (of a field extension $F\rightarrow E$).
5. Algebraic element (of a field extension $F\rightarrow E$).
6. Transcendental element (of a field extension $F\rightarrow E$).
7. $\mathrm{ann}_F(e)$ (the annihilator of $e$ over $F$).
8. $\mathrm{irr}(e,F)$ (the minimal polynomial of $e$ over $F$).
9. $\mathrm{deg}(e,F)$ (the degree of $e$ over $F$; we did not discuss this in class, but it is part of Definition 29.14 in the text).

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

1. Theorem concerning the annihilator of an element ("$\mathrm{ann}_F(e)$ is always an...")

Solve the following problems:

1. Section 29, problems 1, 3, 5, 7, 9, 10, and 14.

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