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 $\mathbb{Z}$-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 $\mathbb{Z}$: Could not fine at all, in my notes or book?

List of prime ideals of F[x]:Could not fine at all, in my notes or book?

List of maximal ideals of F[x]: Let R be a commutative ring with unity. Then M is a maximal ideal of R if and only if R/M is a field.

Theorem characterizing when the quotient ring F[x]/⟨m⟩ will be a field: Let p(x) be an irreducible polynomial in F[x]. If p(x) divides $r(x)s(x)$ for $r(x)$, $s(x) \in F[x]$, then either p(x) divides r(x) or p(x) divides s(x).

So let recap my question here is are the theorems that I listed here are they correct? As well as can you tell me where in the book I might find the, List of maximal ideals of Z and the List of prime ideals of F[x]: I would greatly appreciate the help.

The lists of prime and maximal ideals of $\mathbb{Z}$ are Examples 27.10 and 27.12. The list of maximal ideals of $F[x]$ is Theorem 27.25. Finally, Exercise 27.30 states that the prime ideals of $F[x]$ are the maximal ideals, together with $\left\langle0\right\rangle$. -Steven.Jackson (talk) 10:17, 6 April 2017 (EDT)

Solutions

1. List of prime ideals of $\mathbb{Z}$: $\left\langle0\right\rangle$ and $\left\langle p\right\rangle$ for $p$ prime.
2. List of maximal ideals of $\mathbb{Z}$: $\left\langle p\right\rangle$ for $p$ prime.
3. List of prime ideals of $F[x]$: $\left\langle0\right\rangle$ and $\left\langle f\right\rangle$ for $f$ irreducible.
4. List of maximal ideals of $F[x]$: $\left\langle f\right\rangle$ for $f$ irreducible.
5. Theorem characterizing when the quotient ring $F[x]/\left\langle m\right\rangle$ will be a field: $F[x]/\left\langle m\right\rangle$ is a field if and only if $m$ is irreducible.