[Merged by Bors] - feat(Algebra): transcendence degree is well-defined - #23539
Closed
alreadydone wants to merge 48 commits into
Closed
[Merged by Bors] - feat(Algebra): transcendence degree is well-defined#23539alreadydone wants to merge 48 commits into
alreadydone wants to merge 48 commits into
Conversation
alreadydone
removed
the
blocked-by-other-PR
Collaborator
|
This PR/issue depends on:
|
erdOne
reviewed
Apr 12, 2025
|
🚀 Pull request has been placed on the maintainer queue by erdOne. |
github-actions
Bot
added
the
maintainer-merge
riccardobrasca
approved these changes
Apr 22, 2025
Contributor
|
✌️ alreadydone can now approve this pull request. To approve and merge a pull request, simply reply with |
ghost
added
delegated
alreadydone
added
the
auto-merge-after-CI
|
As this PR is labelled bors merge |
mathlib-bors Bot
pushed a commit
that referenced
this pull request
Apr 22, 2025
In **AlgebraicIndependent/TranscendenceBasis.lean**, we show: + The AlgebraicIndependent sets in a domain form a Matroid, which is used to show all transcendence bases have the same cardinality. Provide algebraic characterizations of the matroid notions `Indep`, `Base`, `cRank`, `Basis`, `closure`, `Flat`, and `Spanning`. + Transcendence bases are exactly algebraic independent families that are spanning (i.e. the whole algebra is algebraic over the algebra generated by the family). On the other hand, a spanning set in a domain contains a transcendence basis. + There always exists an transcendence basis between an arbitrary AlgebraicIndependent set and a spanning set in a domain. + `trdeg R S + trdeg S A = trdeg R A` in a domain, which depends on `AlgebraicIndependent/IsTranscendenceBasis.sumElim_comp`. + A finite AlgebraicIndependent family or spanning family in a domain that has the same cardinality as `trdeg` must be a transcendence basis.
Contributor
|
Pull request successfully merged into master. Build succeeded: |
mathlib-bors
Bot
changed the title
tannerduve
pushed a commit
that referenced
this pull request
May 13, 2025
In **AlgebraicIndependent/TranscendenceBasis.lean**, we show: + The AlgebraicIndependent sets in a domain form a Matroid, which is used to show all transcendence bases have the same cardinality. Provide algebraic characterizations of the matroid notions `Indep`, `Base`, `cRank`, `Basis`, `closure`, `Flat`, and `Spanning`. + Transcendence bases are exactly algebraic independent families that are spanning (i.e. the whole algebra is algebraic over the algebra generated by the family). On the other hand, a spanning set in a domain contains a transcendence basis. + There always exists an transcendence basis between an arbitrary AlgebraicIndependent set and a spanning set in a domain. + `trdeg R S + trdeg S A = trdeg R A` in a domain, which depends on `AlgebraicIndependent/IsTranscendenceBasis.sumElim_comp`. + A finite AlgebraicIndependent family or spanning family in a domain that has the same cardinality as `trdeg` must be a transcendence basis.
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
4 participants
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
In AlgebraicIndependent/TranscendenceBasis.lean, we show:
The AlgebraicIndependent sets in a domain form a Matroid, which is used to show all transcendence bases have the same cardinality. Provide algebraic characterizations of the matroid notions
Indep,Base,cRank,Basis,closure,Flat, andSpanning.Transcendence bases are exactly algebraic independent families that are spanning (i.e. the whole algebra is algebraic over the algebra generated by the family). On the other hand, a spanning set in a domain contains a transcendence basis.
There always exists an transcendence basis between an arbitrary AlgebraicIndependent set and a spanning set in a domain.
trdeg R S + trdeg S A = trdeg R Ain a domain, which depends onAlgebraicIndependent/IsTranscendenceBasis.sumElim_comp.A finite AlgebraicIndependent family or spanning family in a domain that has the same cardinality as
trdegmust be a transcendence basis.