Math 361, Spring 2017, Assignment 8

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

1. Field extension.
2. Extension field.
3. Base field.
4. Algebraic element (of an extension).
5. Transcendental element (of an extension).
6. Minimal polynomial (of an algebraic element).
7. Degree (of an algebraic element).
8. Subextension.
9. Subextension generated by a subset.
10. Finitely generated extension.
11. Simple extension.

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

1. Kronecker's theorem.
2. Theorem concerning irreducibility of minimal polynomials.
3. Classification of simple extensions.

Solve the following problems:

1. Section 29, problems 1, 3, 5, 7, 9, 11, 13, 15, 17, and 18.
2. Describe the elements of $\mathbb{Q}(\sqrt{2})$. Write down two "typical" elements and show how to add, multiply, and invert them.
3. Describe the elements of $\mathbb{Q}(\pi)$. Write down two "typical" elements and show how to add, multiply, and invert them.
4. ("The equation is its own solution.") Show that the equation $x^4+5x^2+6=0$ has no solutions in $\mathbb{Q}$. Then construct an extension of $\mathbb{Q}$ in which this equation does have a solution. Write down two "typical" elements of your extension and show how to add, multiply, and invert them. Then solve the original equation by completely factoring the left hand side over your extension. Do you feel like you are "finished" with the equation? What would make your solution feel more complete?

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