Math 361, Spring 2018, Assignment 5

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

1. Maximal ideal.

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

1. Rational root theorem.
2. Eisenstein's criterion.
3. Theorem relating maximal ideals to fields.
4. Description of the maximal ideals of $\mathbb{Z}$.
5. Description of the maximal ideals of $F[x]$.
6. Theorem characterizing when $F[x]/\left\langle m\right\rangle$ will be a field.

Solve the following problems:

1. Section 23, problems 14, 15, 16, 19, 20, and 21.
2. Is the ring $\mathbb{Q}[x]/\left\langle x^2-2\right\rangle$ a field? Why or why not?
3. Is the ring $\mathbb{R}[x]/\left\langle x^2-2\right\rangle$ a field? Why or why not?

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