I am studying splitting fields, but have realized that I have a conceptual gap when it comes to field automorphisms. More specifically, if $F$ is a field and $\overline{F}$ is its algebraic closure so that $F\leq E\leq\overline{F}$ where $E$ is a finite algebraic field extension. I understand that there are only a finite number of automorphisms of $E$ leaving $F$ fixed, but if we only look at $F$, is there a finite amount of automorphisms just over $F$ as well without thinking about any extensions. Is there any sense to studying field automorphisms unless that specific field is a finite algebraic extension of some other field?
2026-03-31 22:12:17.1774995137
Understanding field automorphisms
39 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in ABSTRACT-ALGEBRA
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