I am studying the notion of a Galois group, and a remark in my notes is the following:

"$L:K$ an extension. Let $H$ be a subgroup of $Aut(L)$. Let $L^{H}$ be the fixed field of $H$.

If $L \leq G(L:K)$, then $L^{H}$ is an intermediate field of $L:K$. "

Now, I don't understand how we can have $L \leq G(L:K)$. $L$ is a an arbitrary field, but the group $G(L:K)$ is a group of $K$-automorphisms of $L$. How can the entire field $L$ be subgroup of a group of automorphisms of $L$?

This seems totally strange to me, as automorphisms are in general different objects to elements of $L$ so how can we even talk about them being subgroups of each other?

  • 5
    $\begingroup$ It's probably a typo and they meant to say "If $H \leq G(L : K)$, then $L^H$ is an intermediate field of $L : K$." $\endgroup$
    – D_S
    Apr 10 '20 at 18:33
  • $\begingroup$ Yeah this is what I thought. Thanks! $\endgroup$
    – Natasha
    Apr 10 '20 at 19:09

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