# Thread: Field problem with factor ring

1. ## Field problem with factor ring

Show that $R[x] / $ is a field.

My proof so far:

Now, $R[x] / = \{ ax+b+ : a,b \in R \}$

Now, [tex](1+<x^2+1>)[tex] is the unity of $R[x] / $

But I have trouble trying to prove every element has a unit.

Let $(ux+v+)$ be in $R[x] / $

I need to find the inverse.

Thank you

2. It happens to be isomorphic to $\mathbb{C}$ just in case you are interested.

Did you do the following theorems?

1.Given a commutative ring $R$ (with unity) and a maximal ideal $M$ then $R/M$ is a field.

2.If $f(x)$ is a non-constant polynomial which is irreducible over the field then $\left< f(x) \right>$ is a maximal ideal of $F[x]$.

So it remains to show that $x^2+1$ is irreducible over $\mathbb{R}$ which it is because it is a degree 2 polynomials with no real zeros.