Math 361, Spring 2019, Assignment 12

Read:

1. Section 31.

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

1. Subspace (of a given vector space).
2. Subspace generated by a subset.
3. Linear combination (from a set $S$).
4. Span (of a set $S$).
5. Linear relation (from a set $S$).
6. Trivial linear relation.
7. Linearly independent (set $S$, in a vector space $V$).
8. Basis (for a vector space $V$).
9. $\mathrm{dim}_F V$ (the dimension of $V$ over $F$).

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

1. Theorem concerning the existence of bases.
2. Theorem concerning relating the cardinalities of two bases for the same vector space.
3. Theorem concerning $\mathrm{dim}_F F[x]/\left\langle m\right\rangle$.
4. Explicit $F$-basis for $F[x]/\left\langle m\right\rangle$ (consisting of powers of the generator $\alpha$).
5. Dimension formula (relating $[E:F], [E:K],$ and $[K:F]$).
6. Explicit basis for $F\rightarrow E$, constructed from bases for $F\rightarrow K$ and $K\rightarrow E$.
7. Theorem relating algebraic elements to finite-dimensional subextensions.

Solve the following problems:

1. Section 30, problems 1, 5, 7, 9, and 10.
2. Section 31, problems 1, 3, 5, 7, 9, and 11.
3. Prove that the infinite set $\{1,x,x^2,x^3,\dots\}$ is a basis for $F[x]$, regarded as a vector space over $F$.
4. Recall that $\overline{\mathbb{Q}}$ denotes the set of all complex numbers that are algebraic over $\mathbb{Q}$. We will prove next week that $\overline{\mathbb{Q}}$ is a field, so that $\mathbb{Q}\rightarrow\overline{\mathbb{Q}}$ is a field extension (for now, you may assume this without proof). Prove that $\overline{\mathbb{Q}}$ is infinite-dimensional over $\mathbb{Q}$, even though every individual element is contained in a finite-dimensional subextension. (Hint: to show that it is infinite-dimensional, it suffices to exhibit finite-dimensional subextensions of arbitrarily large dimension.)
5. If the Axiom of Choice is true, then $\overline{\mathbb{Q}}$, regarded as a vector space over $\mathbb{Q}$, must have a basis. Meditate on what such a basis would need to be like. Is the Axiom of Choice "self-evident?"

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