Math 361, Spring 2018, Assignment 9

Read:

1. Section 30.
2. Section 31.

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

1. Vector space (over an arbitrary field $F$).
2. Subspace.
3. Homomorphism (between two vector spaces; a.k.a. linear transformation).
4. Monomorphism.
5. Epimorphism.
6. Isomorphism.
7. Image (of a linear transformation).
8. Kernel (of a linear transformation).
9. Linear combination (of elements of some set $S$ in some vector space $V$).
10. Span (of some set $S$ in some vector space $V$).
11. Linear relation (among elements of $S$).
12. Trivial linear relation.
13. Linearly independent set.
14. Basis.
15. Dimension.
16. $[E:F]$.

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

1. Theorem concerning the existence of bases.
2. Theorem concerning extension of linearly independent sets to bases.
3. Theorem concerning refinement of spanning sets to bases.
4. Theorem concerning cardinalities of two bases for the same space.
5. Formula for $[F(\beta):F]$ when $\beta$ is algebraic over $F$.
6. Formula for $[F(\beta):F]$ when $\beta$ is transcendental over $F$.
7. Theorem concerning transcendental elements of finite-dimensional extensions.
8. Theorem constraining the order of a finite field.
9. Theorem concerning the existence of fields of order $p^d$ (we did not prove this yet, but we stated it).
10. Theorem relating finite fields with equal cardinalities (we did not prove this yet).
11. Theorem concerning the number of elements required to generate a finite field over $\mathbb{Z}_p$ (we did not prove this yet).
12. Corollary concerning the degrees of irreducible polynomials in $\mathbb{Z}_p[x]$.
13. Dimension formula (relating $[E:F]$ to $[E:K]$ and $[K:F]$).

Solve the following problems:

1. Section 30, problems 1, 5, 7, 8, and 9.
2. Section 31, problems 1, 3, 5, and 7.
3. Construct the field with 125 elements. You do not need to make full addition and multiplication tables; instead, write down two fairly typical elements of your field, and show how to add and multiply them. (Hint: you do not need to run the Sieve of Eratosthenes. How else can you tell whether a cubic polynomial is irreducible?)
4. Suppose that $F\rightarrow E$ is any finite-dimensional extension, and suppose that $K$ is any subextension. Prove that $[K:F]$ is a divisor of $[E:F]$.
5. Prove that $\sqrt{2}\not\in\mathbb{Q}\left(\sqrt[3]{2}\right)$. (Hint: if it were, then $\mathbb{Q}\left(\sqrt{2}\right)$ would be a subextension of $\mathbb{Q}\left(\sqrt[3]{2}\right)$.)

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