Just after my comments on the previous section come the answer to my question. Ends up that “Every nonconstant polynomial in has a root in .”
The last corollary says “Every polynomial f(x) of odd degree in has a root in .” But what about even degree polynomials. It isn’t hard to see, for low even degree polynomials, that there are roots. For what even degree polynomials will there be a root in ?