Originally Posted by

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

x = $\displaystyle 1/\sqrt{(2+2\sqrt2)}+i\sqrt{(1+\sqrt2)/2}$

Show that F contains the element

y = $\displaystyle 1/\sqrt{(2+2\sqrt2)}-i\sqrt{(1+\sqrt2)/2}$

By considering the fact that F must contain $\displaystyle x^2$ we can find that F contains $\displaystyle 1+i$, and hence $\displaystyle i$, 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).