Math 361, Spring 2014, Assignment 2

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

1. Maximal ideal.
2. Prime ideal.
3. Prime subfield (of a given field).

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

1. Theorem relating maximal ideals to fields.
2. Theorem relating prime ideals to integral domains.
3. Theorem relating maximal ideals to prime ideals.
4. Classification of ideals in $F[x]$ (where $F$ is a field).
5. Classification of maximal ideals in $F[x]$.

Solve the following problems:

1. Section 27, problems 3, 5, 16, 18, and 19.
2. Construct a field with exactly four elements. Give names to the elements, then write addition and multiplication tables. Verify that your structure is a field by identifying the multiplicative inverse of each non-zero element. (Hint: begin by finding a quadratic irreducible polynomial in $\mathbb{Z}_2[x]$.)
3. Describe (but do not carry out) constructions producing fields of order 8, 9, 16, 25, and 27. For which integers $n$ do you think your methods can produce fields of order $n$?

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