I have a ring $F =\Bbb Q[x]/(x^2+x+1)$ and a set $E = \{a + bw + cw^2: a,b,c \in \Bbb Q \} \subseteq \Bbb C$. $w = \frac{-1}{2} + \frac{ \sqrt 3}{2}i$. I need to show that these are isomorphic.
So I define a function $f: E \to F$. I need to show first that every $\forall x,y \in E, f(x+y) = f(x) + f(y)$ and $f(xy) = f(x) f(y) $. Then prove that $f$ a bijection.
So $x+y = (a+a') + (b+b')w + (c+c)w^2$ and $xy = (a + bw + cw^2)(a' + b'w + c'w^2)$. The question here for me is what are $f(x+y)$ and $f(xy)$? Is it just the same polynomial because of how $F$ is defined to be a ring of polynomials over the rationals?
I'm a little confused as to how to proceed. Any help is appreciated.
The map $f: \mathbb{Q}[x] \rightarrow F$ given by $f : x \mapsto \omega$ clearly defines a surjective homomorphism. It's kernel is the ideal generated by $x^2+x+1$.