[Merged by Bors] - feat: Add some equivalent characterisations of primitive elements in finite extensions of fields #8609
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Thanks! bors d+ |
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bors r+ |
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…finite extensions of fields (#8609) See the [Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Characterization.20of.20primitive.20elements.20in.20field.20extensions/near/403369086)
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+ generalize `image_rootSet`, `adjoin_rootSet_eq_range` and `splits_comp_of_splits` in Data/Polynomial/Splits and use the last one to golf `splits_of_algHom`, `splits_of_isScalarTower` (introduced in # 8609). + add three new lemmas `mem_range_x_of_minpoly_splits` to simplify the construction of `IntermediateField.algHomEquivAlgHomOfIsAlgClosed` and `Algebra.IsAlgebraic.algHomEquivAlgHomOfIsAlgClosed`, remove the `IsAlgClosed` condition and rename. ~~They could be moved to an earlier file but I refrain from doing that. (#find_home says it's already in the right place)~~ + golf `primitive_element_iff_algHom_eq_of_eval` from # 8609, using a new lemma `IsIntegral.minpoly_splits_tower_top` for the last step. + make `integralClosure_algEquiv_restrict` (from # 8714) computable and rename to `AlgEquiv.mapIntegralClosure` to follow camelCase naming convention and enable dot notation. Co-authored-by: Xavier-François Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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…finite extensions of fields (#8609) See the [Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Characterization.20of.20primitive.20elements.20in.20field.20extensions/near/403369086)
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+ generalize `image_rootSet`, `adjoin_rootSet_eq_range` and `splits_comp_of_splits` in Data/Polynomial/Splits and use the last one to golf `splits_of_algHom`, `splits_of_isScalarTower` (introduced in # 8609). + add three new lemmas `mem_range_x_of_minpoly_splits` to simplify the construction of `IntermediateField.algHomEquivAlgHomOfIsAlgClosed` and `Algebra.IsAlgebraic.algHomEquivAlgHomOfIsAlgClosed`, remove the `IsAlgClosed` condition and rename. ~~They could be moved to an earlier file but I refrain from doing that. (#find_home says it's already in the right place)~~ + golf `primitive_element_iff_algHom_eq_of_eval` from # 8609, using a new lemma `IsIntegral.minpoly_splits_tower_top` for the last step. + make `integralClosure_algEquiv_restrict` (from # 8714) computable and rename to `AlgEquiv.mapIntegralClosure` to follow camelCase naming convention and enable dot notation. Co-authored-by: Xavier-François Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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