Identify quotient ring $\mathbb{R}[x]/(x^2-k), k>0$ I need to identify $\mathbb{R}[x]/(x^2-k)$, where $k>0$ (if $k<0$ I believe it's isomorphic to $\mathbb{C}$).
If we let $f(x) = x^2-k$, then according to Artin, since $\sqrt{k}$ satisfies $f(x)=0$ the set $(1, \sqrt{k})$ is a basis for $\mathbb{R}[\sqrt{k}]$ = $\mathbb{R}[x]/(f)$, i.e. every element of the quotient ring can be written uniquely as $a + b\sqrt{k}$. But this isn't true, because the representation isn't unique.
If we assume $\mathbb{R}[x]/(f) = \mathbb{R}[\sqrt{k}] = \{a + b\sqrt{k} | a,b\in\mathbb{R}\}$ then it looks isomorphic to $\mathbb{R}$, which I don't think it is.
Where have I gone wrong?

More generally, if you quotient by a reducible square-free quadratic polynomial, you get an instance of a ring that I was (once) surprised to find having a name, the "split complex numbers". One you have determined the idempotents in the ring, you find that it is also decomposes as product ring $\Bbb R\times\Bbb R$.
