Math 361, Spring 2016, Assignment 8

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

1. ($F$-)linear combination (of elements of a vector space $V$ with scalars from $F$).
2. ($F$-)span (of a subset of a vector space).
3. ($F$-)linear relation (among elements of $V$).
4. ($F$-)linearly independent set.
5. ($F$-)basis.
6. Dimension (of a vector space over $F$).
7. Dimension (of a field extension $F\rightarrow E$).
8. Degree (of a field extension).
9. Degree (of an algebraic element).

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

1. Theorem concerning extension of linearly independent sets to bases.
2. Theorem concerning refinement of spanning sets to bases.
3. Theorem relating the cardinalities of two bases for the same vector space.
4. Theorem concerning the orders of finite fields.
5. Theorem relating algebraic elements to finite-dimensional subextensions.
6. Dimension theorem.

Solve the following problems:

1. Section 30, problems 1, 3, 5, 7, and 9.
2. Let $F$ denote the field $\mathbb{Q}(\sqrt[3]{2})$, the smallest subfield of $\mathbb{R}$ containing $\mathbb{Q}$ and $\sqrt[3]{2}$.

(a) Compute the dimension of $F$ when regarded as a vector space over $\mathbb{Q}$.

(b) Show that the polynomial $x^2-2$ has no roots in $F$. (Hint: use the Dimension Theorem.)

(c) Compute $[\mathbb{Q}(\sqrt{2},\sqrt[3]{2}):\mathbb{Q}]$.

(d) Find an explicit basis for $\mathbb{Q}(\sqrt{2},\sqrt[3]{2})$, regarded as a vector space over $\mathbb{Q}$. Then give several sample calculations in this field.

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Questions:

• Under definitions, #7 seems to be an elaboration or special case of #6, and #8 seems to be just another name for the phenomenon resulting from the two. Am I mistaken, or is consolidation of the definitions acceptable, or would you rather have specific examples illustrating the situations in which each term would arise?