let K be any field and $\sigma$ be an automorphism of of K where F is the fixed field of $\sigma$ and order of $\sigma$ is "s". now to prove is that [K:F] = s.

well if look at Gal(K/F) it contain all powers of $\sigma$ , so s elements are in it namely {$\sigma$, $\sigma^2$, $\sigma^3$,..., $\sigma^s$ = 1}. but also we know |Gal(K/F)| divides [K:F] and equal iff K/F is galois extension.

so my doubt here is first of all why Gal(K/F) is cyclic and second why K/F is Galois? if these two happen, we are through.

i know i am missing something trivial here. it has been a while i did Galois theory, actually this doubt came in non commutative rings. any hints or ideas or cf.

  • $\begingroup$ I'm assume you're implicitly assuming $s<\infty$ based on how you've written the question. Overall, this is basically the definition of fixed fields for automorphisms. Decompose $K/F$ into a separable and purely inseparable extension, get that the inseparable degree is 1, proving separability. Then normality is easy using the primitive element theorem and linear independence of the characters. $\endgroup$ – Adam Hughes Jun 29 '14 at 4:17

Fix $x\in K$, and let $\{x_0, \dots, x_{r-1}\}$ denote the orbit of $x$ under $\sigma$ for some $r\leq s$, where $x_i = \sigma^i(x)$. Then the polynomial $f(X) = \prod_{i < r} (X - x_i)\in F[X]$, since the coefficients of $f$ are symmetric polyomials in the $\sigma^i(x)$. Since all the $x_i\not\in F$ are distinct, $f$ is irreducible. It follows that $K/F$ is Galois (assuming $s$ is finite). Since $F$ is the fixed field of $G = \langle{\sigma}\rangle\subset \text{Gal}(K/F)$, we then have $[K:F] = \#G = s$.

  • $\begingroup$ thanks a lot. nice proof. this means $\{x_0, \dots, x_{s-1}\}$ is the F basis for K. i guessed that correct, but trouble was to see how do they span K. This is something on which if you could suggest something will be great. $\endgroup$ – Bhaskar Vashishth Jun 29 '14 at 4:46
  • $\begingroup$ It's not a basis for an arbitrary $x\in K$ (take $x\in F$, for example), but it follows from counting degrees if, for example, $K = F(x)$. $\endgroup$ – anomaly Jul 3 '14 at 22:22

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