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I am working on this problem and I was wondering if anyone would be able to help me with it.

The problem states:"Let F be a field and let J be an ideal in F[x]. Prove that J is prime if its generator is irreducible over F."

I am not sure what "...irreducible over F" means. I've done research and I keep finding problems related to polynomials being irreducible.

I know that a field is a commutative ring with unity in which every non zero element is invertible. An ideal is a nonempty subset that is closed under addition, negatives and it absorbs products. And I know that prime means that "If ab is in J, then a is in J or b is in J.

Any help is greatly appreciated! Thanks!

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    It means irreducible as an element of the ring $F[x].$ See https://en.wikipedia.org/wiki/Irreducible_element2012-12-13
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    Glad to help!{}2012-12-13

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