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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Questions

List of prime ideals of Z-If R is a ring with unity 1, then the map $\phi :\mathbb{Z} \rightarrow R$ given by $ \phi(n) =n \cdot 1$ for $n \in mathbb{Z} into R$ List of maximal ideals of Z: Could not fine at all? List of prime ideals of F[x]:Could not fine at all? List of maximal ideals of F[x]: Theorem characterizing when the quotient ring F[x]/⟨m⟩ will be a field.

I am not done yet.