Math 361, Spring 2017, Assignment 7

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

1. List of prime ideals of $\mathbb{Z}$.
2. List of maximal ideals of $\mathbb{Z}$.
3. List of prime ideals of $F[x]$.
4. List of maximal ideals of $F[x]$.
5. Theorem characterizing when the quotient ring $F[x]/\langle m\rangle$ will be a field.

Solve the following problems:

1. Section 27, problems 5, 7, and 9.
2. Construct a field with exactly eight elements. Make addition and multiplication tables for your field. Is your field isomorphic to $\mathbb{Z}_8$? (Hint: first find an irreducible cubic in $\mathbb{Z}_2[x]$.)

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