I'm trying to prove $\mathbb C =\mathbb R(a+bi)$, where $(a+bi) \in \mathbb C$ and $b \neq0$
I'm doing like this:
$[\mathbb C:\mathbb R]=[\mathbb C: \mathbb R(a+bi)][\mathbb R(a+bi):\mathbb R]$.
If we prove $[\mathbb R(a+bi): \mathbb R]=2$ we done, since we know $[\mathbb C:\mathbb R]=2$, then $[\mathbb C:\mathbb R(a+bi)]=1$ and $\mathbb R(a+bi) =\mathbb C$.
In order to prove $[\mathbb R(a+bi): \mathbb R]=2$, I'm trying to find the minimal polynomial of (a+bi) over $\mathbb R$. I found a candidate: $p(x)=x^2 -2ax+a^2+b^2$. I know that $p(a+ bi)=0$, but how to prove this polynomial is irreducible? thanks.