Degree of the field extension.

Question

Could you help me showing that for a field $F$ the degree of the extension $F(x^{2}+\frac{1}{x^{2}})\subset F(x)$ is $4$?

I have found the polynomial $y^{4}-(x^{2}+\frac{1}{x^{2}})y^{2}+1$ such that $x$ is its root, but how can I prove that it is the minimal one?

Thanks in advance.

Answer 1

Observe that $F(x^2+\frac{1}{x^2})\subset F(x^2)\subset F(x)$. Now prove that each extension is of degree two.

Answer 2

The solution by Brett is great. I just what to add that you can explicitly write the four automorphisms:

$$x\mapsto x$$ $$x\mapsto -x$$ $$x\mapsto \frac{1}{x}$$ $$x\mapsto -\frac{1}{x}$$