The extension i'm working with is $\mathbb{Q}(x)$ over $\mathbb{Q}$ where $x$ is a primitive $5$-root of unity.
While trying to find the fixed field of the automorphism $f:x\rightarrow x^{4}$ I obtained that its fixed field is only $\mathbb{Q}$ which has order 1 over $\mathbb{Q}$ in contradiction to the fact that such $f$ has order 2.
Let $y=a+b(x)+c(x)^{2}+d(x)^{3}$ then applying the automorphism and knowing that $x^{4}=-x^{3}-x^{2}-x-1$ I got $a=a-b, b=-b, c=-b+d, d=-b+c$ meaning the elements fixed are of the form $y=a+c(x)^{2}+c(x)^{3}$ which I believe is not a field. What am I doing wrong?
The basefield is the cyclotomic field $CF_5$, of dimension $4$ over the rationals $\mathbb{Q}$. The field extension under consideration is $F = CF_5(x+x^4) = CF_5(x^2+x^3)$. To view the degree of this extension take the element $u = x - x^4$ and note that $u \notin F $ and that it satisfies the second degree equation $u^2 - C = 0$, with coefficients in F since $C = 2*(x+x^4)+3*(x^2+x^3)$.