I have a question about the complex numbers given as an extension field.
I know that complex numbers can be seen as $\mathbb{R}\left[x\right]/<x^2+1>$ (Well, I really don't know it yet, because I haven't understood the argument). But I don't understand why is that in Fraleigh they say that this field consists in all elements of the form $a+b\alpha$ where $\alpha=x+<x^2+1>$ and $a,b\in\mathbb{R}$. If I replace in $a+b\alpha$ it becomes: $a+bx+b<x^2+1>$. But i thought the set $\mathbb{R}\left[x\right]/<x^2+1>$ consisted in the elements of the form $\sum_{k=0}^{\infty}a_kx^k+<x^2+1>$ where $a_i=0$ for all but a finite number of values of $i$. How can I show that it's the same? And is it really the same?
I know that $\alpha$ plays the role of $i$, but...