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Prime and Maximum Ideals

I am having difficulty formulating the proof for this problem. I understand the concepts but not sure how to go about the actual proof. Please help! In regards to J being an ideal I proved that no problem so I omitted those details from the problem.

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Proof:
First I show that J is a prime ideal.
For any g(x), h(x) in Z[x] and g(x)h(x) is in J, we have g(5)h(5) = 0. Then g(5) = 0 or h(5) = 0.
If g(5) = 0, then g(x) is in J; if h(5) = 0, then ...

Solution Summary

This article provides an example of proving that something is a prime and maximal ideal.

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