[Merged by Bors] - feat(number_theory): define number fields, function fields and their rings of integers #8701
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Let `L` be a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain. Then `integral_closure A L` is also a Dedekind domain. In combination with the definitions of #8701, we can conclude that rings of integers are Dedekind domains.
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Let `L` be a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain. Then `integral_closure A L` is also a Dedekind domain. In combination with the definitions of #8701, we can conclude that rings of integers are Dedekind domains.
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Let `L` be a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain. Then `integral_closure A L` is also a Dedekind domain. In combination with the definitions of #8701, we can conclude that rings of integers are Dedekind domains.
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The typeclass `is_integral_closure A R B` states `A` is the integral closure of `R` in `B`, i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`. We also show that any integral extension of `R` contained in `B` is contained in `A`, and the integral closure is unique up to isomorphism. This was suggested in the review of #8701, in order to define a characteristic predicate for rings of integers.
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The typeclass `is_integral_closure A R B` states `A` is the integral closure of `R` in `B`, i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`. We also show that any integral extension of `R` contained in `B` is contained in `A`, and the integral closure is unique up to isomorphism. This was suggested in the review of #8701, in order to define a characteristic predicate for rings of integers. Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
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Let `L` be a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain. Then `integral_closure A L` is also a Dedekind domain. In combination with the definitions of #8701, we can conclude that rings of integers are Dedekind domains.
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Co-Authored-By: Eric Wieser <wieser.eric@gmail.com>
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Let `L` be a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain. Then `integral_closure A L` is also a Dedekind domain. In combination with the definitions of #8701, we can conclude that rings of integers are Dedekind domains.
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| -- The `is_number_field K` hypothesis is not used but is required for | ||
| -- this definition to make sense. | ||
| @[nolint unused_arguments] |
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Shouldn't we leave it out anyway? It would be sufficient to assume it for lemmas, right?
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I'm just a bit wary of someone having open function_field number_field somewhere, trying to get the ring of integers of a function field, and receiving the integral closure of ℤ instead. Complaining about a missing instance would prevent that issue. But I guess they will notice the issue as soon as they try to prove anything with it.
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…DD (#8847) Let `L` be a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain. Then any `is_integral_closure C A L` is also a Dedekind domain, in particular for `C := integral_closure A L`. In combination with the definitions of #8701, we can conclude that rings of integers are Dedekind domains.
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…rings of integers (#8701) Co-Authored-By: Ashvni <ashvni.n@gmail.com> Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
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Co-Authored-By: Ashvni ashvni.n@gmail.com
R ≃ R', map a basis forR-moduleMtoR'-moduleM#8699