AlgebraInLean/README.md

Algebra in Lean

A hands-on introduction to formalizing algebra with the proof assistant Lean and its mathematical library Mathlib. The course accompanies the lecture notes Algebra und Zahlentheorie (Stefan Kebekus, Universität Freiburg) and is aimed at students who are learning algebra and have little prior experience with Lean.

Selected proofs from the lecture notes are translated into Lean sentence by sentence: the German original of each sentence is quoted as a comment directly above the Lean code implementing it, so you can see how the standard phrases of a lecture-style proof translate into Lean/Mathlib.

Getting started

1. Install Lean (elan, VS Code and the Lean 4 extension).

2. Clone this repository and fetch the precompiled Mathlib cache:

   git clone <repository url>
   cd AlgebraInLean
   lake exe cache get
   

3. Open the folder in VS Code and start with AlgebraInLean/LittleFermat.lean.

Contents

File	Lecture notes	Topic
AlgebraInLean/Basics.lean	Kapitel 2	First steps: tactics, groups in Mathlib
AlgebraInLean/Quotients.lean	Kapitel 2, 17	Kernels, normal subgroups, quotient groups
AlgebraInLean/Ideals.lean	Kapitel 9	Ideals; ℤ is a principal ideal ring
AlgebraInLean/Polynomials.lean	Kapitel 7	Polynomials; Eisenstein's criterion applied
AlgebraInLean/Degrees.lean	Kapitel 3	Field extensions; the tower law
AlgebraInLean/LittleFermat.lean	Kapitel 17	Fermat's little theorem via Lagrange
AlgebraInLean/Cauchy.lean	Kapitel 18	The key lemma on fixed points and Cauchy's theorem
AlgebraInLean/Frobenius.lean	Kapitel 14	Finite fields; the Frobenius endomorphism
AlgebraInLean/Galois.lean	Kapitel 15, 16	Galois theory; fixed fields
AlgebraInLean/Reciprocity.lean	Kapitel 24	Quadratic reciprocity; computing Legendre symbols

The rendered course notes

The folder Book/ contains the same material as a Verso book (the technical setup follows Peter Pfaffelhuber's leancourse). To render it:

lake build
lake exe algebrainlean --output _out
python3 -m http.server 8000 -d _out/html-multi   # then open http://localhost:8000

All Lean code in the book is elaborated during the build, so the rendered proofs are guaranteed to compile. ./deploy.sh builds the site and copies it to public/, which is synced to the CPLX web server as for the lecture notes.

License

CC-BY 4.0, like the lecture notes.