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!