feat(FunctionField): a function field is finite over Fq(y) for transcendental y

[xgenereux]

Given F a function field over Fq, we know by assumption that it is finite over RatFunc Fq.
This PR adds the result that it is also finite over Fq(y) for any transcendental y over Fq.

Notes:

1. Instance synthesis is known to be slow with IntermediateField, so we guide it in the proof of finiteDimensional_of_adjoin_transcendental.
2. While looking for the proof of Module.Finite.top_left, it took me a while to find Module.Finite.of_surjective as I was looking for equiv lemmas. I've added a comment in the docstring of Module.Finite.of_equiv_equiv to ease discovery.
3. Since LinearMap.id' requires σ to have the same domain and codomain, we explicitly provide a generalized version of it in the proof of Module.Finite.top_left.

Co-authored-by: María Inés de Frutos Fernández <mariaines.dff@gmail.com>

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• depends on: [Merged by Bors] - chore(FunctionField): rename vars when constant field is not nec. finite #37894
• depends on: [Merged by Bors] - chore(RatFunc/Luroth): split file and small changes #37397

[github-actions]

PR summary 044631201f

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.NumberTheory.FunctionField	2244	2253	+9 (+0.40%)

Import changes for all files

Files	Import difference
3 files Mathlib.NumberTheory.ClassNumber.FunctionField Mathlib.NumberTheory.FunctionField Mathlib.NumberTheory.RatFunc.Ostrowski	9

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Declarations diff

+ Algebra.IsAlgebraic.adjoin_algebraMap_X
+ FiniteDimensional.adjoin_X
+ FiniteDimensional.adjoin_algebraMap_X
+ finiteDimensional_of_adjoin_transcendental
+ isAlgebraic_X_over_adjoin_transcendental
+ top_left

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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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).

[MichaelStollBayreuth]

I would greatly appreciate a (separate) PR that changes Fq (which misleadingly suggests that Fq is a finite field) to a more generic field variable like K throughout this file.

[xgenereux]

I've added a dependency to your requested PR.

[MichaelStollBayreuth]

You'll have to merge master to fix the build (see the announcement on Zulip).

[xgenereux]

Thanks for the heads up. (I'll wait on #37894.)

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - chore(FunctionField): rename vars when constant field is not nec. finite #37894
• [Merged by Bors] - chore(RatFunc/Luroth): split file and small changes #37397
  By Dependent Issues (🤖). Happy coding!