Principle Ideals of a Polynomial Quotient Ring

[DeldotB]

Homework Statement

Let A be the algebra $\mathbb{Z}_5[x]/I$ where $I$ is the principle ideal generated by $x^2+4$ and $\mathbb{Z}_5[x]$ is the ring of polynomials modulo 5.

Find all the ideals of A
Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.

Homework Equations

None

The Attempt at a Solution

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I have no idea where to start. Why is $x^2+4$ an ideal? How do I find other ideals?

I have been asked about invertible elements in rings like $\mathbb{Z}/n\mathbb{Z}$ (just the elements co-prime to n) but how does this concepts relate to polynomials? Are invertible elements in polynomial rings also "coprime" in some sense??

Thankyou

 

[andrewkirk]

I have no idea where to start. Why is $x^2+4$ an ideal? How do I find other ideals?

It's not, and the problem didn't say it was. It is the generator of a (principal) ideal.

You won't be able to even get started on this if you don't know what an ideal is and what a principal ideal is. Your text and/or notes will have given you definitions.

What are they?