Math 361, Spring 2015, Assignment 5

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

1. Field extension.
2. Base field.
3. Extension field.
4. Injection (associated with a field extension).
5. Subextension.
6. Trivial extension.
7. Improper subextension.

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

1. Classification of prime ideals and maximal ideals in PIDs.
2. Theorem concerning ideals in $F[x]$ (where $F$ is any field).
3. Enumeration of elements of the field $F[x]/\left\langle p\right\rangle$ (that is, the statement that each element of this field can be expressed uniquely in a certain "preferred" or "standard" form; this statement is not in the book).

Solve the following problems:

1. Section 27, problem 7.
2. Construct a field with exactly four elements. Calculate explicit addition and multiplication tables, and verify that every non-zero element does in fact have a multiplicative inverse. (Hint: start by finding an irreducible quadratic polynomial with coefficients in $\mathbb{Z}_2$.)
3. Suppose you have identified an irreducible polynomial $f$ of degree $d$ over $\mathbb{Z}_p$. How many elements does the field $\mathbb{Z}_p[x]/\left\langle f\right\rangle$ have?

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