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Prove that the ideal is prime in Z[x] but not maximal in

Chapter 17, Problem 32E

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QUESTION:

Prove that the ideal \(\left\langle x^{2}+1\right\rangle\) is prime in Z[x] but not maximal in Z[x].

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QUESTION:

Prove that the ideal \(\left\langle x^{2}+1\right\rangle\) is prime in Z[x] but not maximal in Z[x].

ANSWER:

Step 1 of 2

Assume the following.

Then, we must have the following.

Since  are all integers, we can take:

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