Math 361, Spring 2015, Assignment 6

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

1. Algebraic element (of a field extension).
2. Transcendental element.
3. Minimal polynomial (of an algebraic element).
4. Monic polynomial.
5. $\mathrm{irr}(\alpha, F)$.
6. Algebraic extension.
7. Subextension generated by a subset.
8. Simple extension.

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

1. Classification of simple extensions.

Solve the following problems:

1. Section 29, problems 3, 7, 9, and 11.
2. Let $\mathbb{Q}(\sqrt{2},\sqrt{3})$ denote the smallest subfield of $\mathbb{R}$ containing $\mathbb{Q}$ and $\sqrt{2}$ and $\sqrt{3}$. Show that every element of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ can be written uniquely in the form $a_0+a_1\sqrt{2}+a_2\sqrt{3}+a_3\sqrt{6}$ with $a_i\in\mathbb{Q}$. Try to make addition and multiplication formulas for this field. (Hint: build this field by making two simple extensions in sequence.)

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