@Matt: R[x] refers to the finite polynomials a+bx+cx^2+... This works when R is a ring, and fields are special cases of rings so R can also be a field. With i^2=-1, all polynomials in $i$ reduce to the form $a+bi$. Now for K a field, K(x) refers to the smallest field containing K and x. We have Q[i]=Q(i) because inverses of a+bi can also be written in this form. Something like $\mathbb{Q}[\pi]$ is not a field however, even though $\mathbb{Q}(\pi)$ is.

$K(x)$ can be constructed as the field of fractions of $K[x]$ of course

When $x$ is algebraic we can speak of K(x) and K[x] interchangeable though. Since the norm defined by multiplying automorphisms of an element evaluates to elements in the base field, we can divide the product out by the element we want to invert and exhibit an inverse, same as we do for a+bi in Q[i].