# A Ring Isomorphism Problem

1. Jul 10, 2008

### e(ho0n3

The problem statement, all variables and given/known data
Prove that $$Q[x]/\langle x^2 - 2 \rangle$$ is ring-isomorphic to $$Q[\sqrt{2}] = \{a + b\sqrt{2} \mid a,b \in Q\}$$.

The attempt at a solution
Denote $$\langle x^2 - 2 \rangle$$ by I. $$a_0 + a_1x + \cdots + a_nx^n + I$$ belongs to Q[x]/I. It has n + 1 coefficients which somehow map to a and b. I don't think any injection can do this. I'm stumped. Any hints?

2. Jul 10, 2008

### e(ho0n3

I think I got it: Let f(x) be an element of Q[x]. f(x) may be rewritten as (xx - 2)q(x) + r(x) for some q(x), r(x) in Q[x] with r(x) = 0 or deg r(x) < deg (xx - 2) = 2. Thus, r(x) has the form a + bx, a and b both belonging to Q. Aha!