69 lines
2.8 KiB
Markdown
69 lines
2.8 KiB
Markdown
# Algebra in Lean
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A hands-on introduction to formalizing algebra with the proof assistant
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[Lean](https://lean-lang.org) and its mathematical library
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[Mathlib](https://leanprover-community.github.io). The course accompanies
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the lecture notes *Algebra und Zahlentheorie* (Stefan Kebekus,
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Universität Freiburg) and is aimed at students who are learning algebra
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and have little prior experience with Lean.
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Selected proofs from the lecture notes are translated into Lean
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*sentence by sentence*: the German original of each sentence is quoted
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as a comment directly above the Lean code implementing it, so you can
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see how the standard phrases of a lecture-style proof translate into
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Lean/Mathlib.
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## Getting started
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1. [Install Lean](https://leanprover-community.github.io/get_started.html)
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(`elan`, VS Code and the Lean 4 extension).
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2. Clone this repository and fetch the precompiled Mathlib cache:
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```bash
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git clone <repository url>
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cd AlgebraInLean
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lake exe cache get
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```
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3. Open the folder in VS Code and start with
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`AlgebraInLean/LittleFermat.lean`.
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## Contents
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| File | Lecture notes | Topic |
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|------|---------------|-------|
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| `AlgebraInLean/Basics.lean` | Kapitel 2 | First steps: tactics, groups in Mathlib |
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| `AlgebraInLean/Quotients.lean` | Kapitel 2, 17 | Kernels, normal subgroups, quotient groups |
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| `AlgebraInLean/Ideals.lean` | Kapitel 9 | Ideals; ℤ is a principal ideal ring |
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| `AlgebraInLean/Polynomials.lean` | Kapitel 7 | Polynomials; Eisenstein's criterion applied |
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| `AlgebraInLean/Degrees.lean` | Kapitel 3 | Field extensions; the tower law |
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| `AlgebraInLean/Elements.lean` | Kapitel 3 | The degree of an algebraic element |
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| `AlgebraInLean/Transitivity.lean` | Kapitel 3 | Transitivity of algebraicity |
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| `AlgebraInLean/LittleFermat.lean` | Kapitel 17 | Fermat's little theorem via Lagrange |
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| `AlgebraInLean/Cauchy.lean` | Kapitel 18 | The key lemma on fixed points and Cauchy's theorem |
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| `AlgebraInLean/Frobenius.lean` | Kapitel 14 | Finite fields; the Frobenius endomorphism |
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| `AlgebraInLean/Galois.lean` | Kapitel 15, 16 | Galois theory; fixed fields |
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| `AlgebraInLean/Reciprocity.lean` | Kapitel 24 | Quadratic reciprocity; computing Legendre symbols |
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## The rendered course notes
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The folder `Book/` contains the same material as a
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[Verso](https://github.com/leanprover/verso) book (the technical setup
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follows Peter Pfaffelhuber's
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[leancourse](https://github.com/pfaffelh/leancourse)). To render it:
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```bash
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lake build
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lake exe algebrainlean --output _out
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python3 -m http.server 8000 -d _out/html-multi # then open http://localhost:8000
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```
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All Lean code in the book is elaborated during the build, so the
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rendered proofs are guaranteed to compile. `./deploy.sh` builds the
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site and copies it to `public/`, which is synced to the CPLX web
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server as for the lecture notes.
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## License
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CC-BY 4.0, like the lecture notes.
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