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