What's the dimension of $\mathbb C$ as a vector space over $\mathbb{R}$? What's the dimension of $\mathbb C$ as a vector space over $\mathbb{R}$?
I think that the answer is $2$, because $\mathbb R^2 \cong \mathbb{C}$ (since all complex numbers can be mapped to $\mathbb R^2$ and vice versa).
Is this correct, or is it a trick question (in which case I'm wrong)?

Hint:
$\lbrace 1,i\rbrace$ is a basis. 
As for what you have said, there is an isomorphism $$\frac{\mathbb{R}[x]}{\langle x^2+1\rangle}\simeq \mathbb{C}$$ where $\mathbb{R}[x]$ is the polynomial ring.
