$Show$ that a finite Galois extension has a finite number of intermediate fields.
2026-04-07 13:43:59.1775569439
Show that a finite Galois extension has a finite number of intermediate fields.
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Let $E$ be a finite galois extension of $F$. The intermediate fields are in one-to-one correspondence with the subgroups of $Gal(E/F)$ by the fundamental theorem. This group has finite order and therefore finitely many subgroups as it has finitely many subsets so that the number of intermediate fields is finite.