1. The problem statement, all variables and given/known data I am just learning about field extensions, so I am not as comfortable with them yet as I'd like to be. I have a more general homework problem, but let me post a special case of the part I am focusing on which contains the part that is confusing me. Suppose I have an extension K/Q of degree 2, where Q is the rationals. Also suppose now that I have an irreducible quadratic f(x) in Q[x] with one root in K. From the problem as is, I am supposed to find such a K and f(x) for which there is only one root in K. But from the quadratic formula it seems to me that as soon as one root is in K, the other must be as well. What am I missing?