feat(FieldTheory): basic theories about transcendental separable extension

[Thmoas-Guan]

The formalization of [Stacks 030W] via [Stacks 030U].

---

• depends on: feat(FieldTheory): definition of transcendental separable field extension #37934

[mathlib-bors]

This pull request is now in draft mode. No active bors state needed cleanup.

While this PR remains draft, bors will ignore commands on this PR. Mark it ready for review before using commands like bors r+ or bors try.

[github-actions]

PR summary 1a3500f1bf

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.FieldTheory.TranscendentalSeparable (new file)	1997

---

Declarations diff

+ Algebra.IsSeparablyGenerated
+ Algebra.IsTranscendentalSeparable
+ Algebra.isSeparablyGenerated_of_charZero
+ Algebra.isSeparablyGenerated_of_equiv
+ Algebra.isTranscendentalSeparable_iff_isSeparablyGenerated_of_essFiniteType
+ Algebra.isTranscendentalSeparable_of_charZero
+ Algebra.isTranscendentalSeparable_of_isSeparablyGenerated
+ Algebra.isTranscendentalSeparable_of_perfectField
+ Algebra.isTranscendentalSeparable_tfae
+ IsReduced.tensorProduct_of_forall_fg_intermediateField
+ adjoinPthRoots
+ adjoinPthRootsPthRoot
+ adjoinPthRootsPthRoot_bijective
+ adjoinPthRootsPthRoot_pow
+ adjoinPthRootsSelf
+ adjoinPthRootsSelf_algebraMap
+ adjoinPthRoots_pth_power_mem_bot
+ instance : Algebra k (adjoinPthRoots k p) := (frobenius k p).toAlgebra
+ instance : ExpChar (AlgebraicClosure k) p := ExpChar.of_injective_algebraMap' k _
+ instance : Field (adjoinPthRoots k p) := inferInstanceAs (Field k)
+ instance [Fact (Nat.Prime p)] : Algebra.IsAlgebraic k (adjoinPthRoots k p)
+ isReduced_injective_to_prod_localizations
+ isReduced_of_quotient_separable
+ isReduced_of_quotient_separable_of_field
+ linearIndepOn_pow_of_isReduced_tensorProduct
+ localization_minimal_isField
+ polynomialTensorProductEquiv
+ polynomialTensorProductEquiv_map_algebraMap
+ quotientPolynomialTensorProductEquiv
+ tensorProduct_isReduced_of_isSeparablyGenerated_isDomain
+ tensorProduct_isReduced_of_isSeparablyGenerated_of_isReduced
+ tensorProduct_isReduced_of_isTranscendentalBasis_of_isDomain
+ tensorProduct_isReduced_of_isTranscendentalBasis_of_isReduced
+ tensorProduct_isReduced_of_isTranscendentalSeparable_of_isReduced
+ tensorProduct_isReduced_of_isTranscendentalSeparable_of_isReduced_of_essFiniteType
+ toLocalizationMinimal

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.

---

No changes to technical debt.

You can run this locally as

./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).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(FieldTheory): definition of transcendental separable field extension #37934
  By Dependent Issues (🤖). Happy coding!