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[Merged by Bors] - feat(number_theory/kummer_dedekind): define conductor and add basic API plus theorem #16833

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[Merged by Bors] - feat(number_theory/kummer_dedekind): define conductor and add basic API plus theorem #16833

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@Paul-Lez Paul-Lez commented Oct 6, 2022 •
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In this PR we define the conductor ideal and prove some basic results about it.
In a later PR (#16870), these will be used to prove the Dedekind-Kummer theorem in full generality.

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@Paul-Lez Paul-Lez added the WIP Work in progress label Oct 6, 2022
@Paul-Lez Paul-Lez added awaiting-review The author would like community review of the PR awaiting-CI The author would like to see what CI has to say before doing more work. and removed WIP Work in progress labels Oct 7, 2022
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alreadydone
alreadydone reviewed Oct 13, 2022
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These are mostly superficial comments without actually opening the branch in VSCode, and I may or may not take another look later ...

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riccardobrasca
riccardobrasca approved these changes Oct 14, 2022
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open ideal polynomial double_quot unique_factorization_monoid algebra ring_hom

local notation R `[` x `]` := adjoin R ({x} : set S)
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I am not sure, but since we now have R[X] for polynomials (even if it's not opened here), this can be a little confusing. But I agree we need a good notation for adjoin.

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I'll try and find something else!

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@alreadydone alreadydone Oct 14, 2022 •
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Note that intermediate_field.adjoin has this weird Unicode notation.

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/-- The canonical morphism of rings from `R[x] ⧸ (I*R[x])` to `S ⧸ (I*S)` is an isomorphism
when `I` and `(conductor R x) ∩ R` are coprime. -/
noncomputable def quot_adjoin_equiv_quot_map (hx : (conductor R x).comap (algebra_map R S) ⊔ I = ⊤)
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Maybe a simps tag? It depends on what you want to do with this map, but we probably want at least a minimal API around it.

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@Paul-Lez Paul-Lez Oct 16, 2022 •
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The I tried adding simps apply but it gave the "wrong" projection lemma so I added a projection lemma manually.

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@Paul-Lez Paul-Lez Oct 16, 2022 •
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In a later PR - possibly #16870 - I'll prove the usual criterion for the Dedekind-Kummer theorem in terms of the divisibility of the index [O_K : Z[a] ] by the prime ideal I so I'll probably end up adding more API to do that.

Paul-Lez and others added 2 commits October 16, 2022 13:02
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Change notation and add simp lemma
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Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com>
@Paul-Lez Paul-Lez added t-algebra Algebra (groups, rings, fields etc) t-number-theory Number theory (also use t-algebra or t-analysis to specialize) labels Oct 16, 2022
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Thanks!

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@github-actions github-actions bot added ready-to-merge All that is left is for bors to build and merge this PR. (Remember you need to say `bors r+`.) and removed awaiting-review The author would like community review of the PR labels Oct 19, 2022
bors bot pushed a commit that referenced this pull request Oct 19, 2022
@Paul-Lez
feat(number_theory/kummer_dedekind): define conductor and add basic A…
813725f 
…PI plus theorem (#16833)

In this PR we define the conductor ideal and prove some basic results about it. 
In a later PR (#16870), these will be used to prove the Dedekind-Kummer theorem in full generality.



Co-authored-by: Paul Lezeau <paul.lezeau@gmail.com>
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Pull request successfully merged into master.

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@bors bors bot changed the title feat(number_theory/kummer_dedekind): define conductor and add basic API plus theorem [Merged by Bors] - feat(number_theory/kummer_dedekind): define conductor and add basic API plus theorem Oct 19, 2022
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@bors bors bot deleted the test_branch branch October 19, 2022 09:54
@eric-wieser eric-wieser added the hacktoberfest-accepted Without this label hacktoberfest is scared off by bors label Oct 29, 2022
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