Show that $\displaystyle \sqrt {3} $ is not in $\displaystyle \mathbb {Q} [ ^4 \sqrt {2} ] $

Proof so far:

Let $\displaystyle x = ^4 \sqrt {2} $

Assume to the opposite that $\displaystyle \sqrt {3} = a + bx+cx^2 +dx^3 \ , \ a,b,c,d \in \mathbb {Q} $

Now, I want to take the trace of both side, but I'm really having problem with field trace, I don't know how to take the trace of neither side, are there any references I can find online that reteach me what trace is and how to compute it?

Thank you very much!