Say we have a field F and such that Q is a subfield of F and additionally F contains the element

x =

Show that F contains the element

y =

By considering the fact that F must contain

we can find that F contains

, and hence

, and various other results like this. Can we show using various combinations of these results that F contains y? If so, then I'll be able to use this to answer a question about splitting fields.

(I want to find [Q(x,y):Q], and clearly this will just be [Q(x),Q] if y is in Q(x) which will simplify the problem).