feat(RingTheory): totally split algebras over a field

[chrisflav]

We show that for a totally split algebra over a field, the k-rational points are in bijection with the prime spectrum. We also show that an étale algebra over a separably closed field is totally split.

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[github-actions]

PR summary 3325a31dc8

Import changes exceeding 2%

%	File
+3.61%	Mathlib.RingTheory.TotallySplit

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.TotallySplit	2244	2325	+81 (+3.61%)

Import changes for all files

Files	Import difference
Mathlib.RingTheory.TotallySplit	81

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Declarations diff

+ algHomEquivPrimeSpectrum
+ algebraMap_bijective
+ bijective_algebraMap_quotient
+ coe_evalAlgHom
+ equivPiOfIsSepClosed
+ equivPiOfIsSepClosed_comap
+ equivPiOfIsSepClosed_self_apply
+ equivPi_apply
+ instance : Etale R R
+ instance : FormallyEtale R R := of_formallyUnramified_and_formallySmooth
+ instance [EssFiniteType K A] [FormallyEtale K A] (p : Ideal A) [p.IsPrime] :
+ instance [IsSepClosed k] [EssFiniteType k R] [FormallyEtale k R] : IsFiniteSplit k R := by

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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Increase in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
6217	2	backward.isDefEq.respectTransparency

Current commit fc3eeae02b
Reference commit 3325a31dc8

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).