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doc(FieldTheory/SeparableDegree): fix the formula in docstring (#9318)
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acmepjz committed Dec 28, 2023
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7 changes: 4 additions & 3 deletions Mathlib/FieldTheory/SeparableDegree.lean
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Expand Up @@ -25,9 +25,10 @@ This file contains basics about the separable degree of a field extension.
**Remark:** if `E / F` is not algebraic, then this definition makes no mathematical sense,
and if it is infinite, then its cardinality doesn't behave as expected (namely, not equal to the
field extension degree of `separableClosure F E / F`). For example,
$\operatorname{Emb}_{\mathbb{Q}}(\mathbb{Q}(\mu_{p^\infty}))\cong\mathbb{Z}_p^\times$ which is
uncountable, while $[\mathbb{Q}(\mu_{p^\infty}):\mathbb{Q}]$ is countable.
field extension degree of `separableClosure F E / F`). For example, if $ F = \mathbb{Q} $ and
$ E = \mathbb{Q}( \mu_{p^\infty} ) $, then $ \operatorname{Emb}_F (E) $ is in bijection with
$\operatorname{Gal}(E/F)$, which is isomorphic to
$ \mathbb{Z}_p^\times $, which is uncountable, while $ [E:F] $ is countable.
**TODO:** prove or disprove that if `E / F` is algebraic and `Emb F E` is infinite, then
`Field.Emb F E` has cardinality `2 ^ Module.rank F (separableClosure F E)`.
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