Math 361, Spring 2014, Assignment 5

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

1. Vector space (over a field $F$).
2. Spanning set.
3. Linearly independent set.
4. Basis.
5. Dimension.
6. Algebraic extension.
7. Algebraically closed field.
8. Algebraic closure.

Carefully state the following theorems:

1. Theorem concerning the existence of bases.
2. Dimension formula.
3. Theorem relating finite-dimensional extensions to algebraic extensions.
4. Theorem concerning the existence of algebraic closures.

Solve the following problems:

1. Section 30, problems 1, 4, 5, 7, and 9.
2. Section 31, problems 1, 3, 5, 23, and 24.

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