1. ## Field extension problem

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

Proof so far:

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

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

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!

2. You can prove this by showing, $\sqrt{3} = a+b\sqrt[4]{2}+c\sqrt[4]{4}+d\sqrt[4]{8}$ is impossible by squaring sides.

3. thanks, but it is just that I want to learn more about field trace, I'm really really lost in this thing.