1
Setup
▶
1.1
The fields and the primes
1.2
The Teichmüller character
1.3
The additive character
1.4
Primes above \(p\)
2
The Gauss sum
▶
2.1
Gauss and Jacobi sums
2.2
Congruences
2.3
Galois action
3
The valuation function \(s\)
▶
3.1
General properties
3.2
Upper bound by the sum of digits
3.3
The digit formula
4
Ideal factorization of Gauss sums
▶
4.1
Setup for Stickelberger’s theorem
4.2
Descent of \(g_d^m\) to \(k\)
4.3
Factorization in \(L\)
4.4
Primes of \(k\) above \(p\)
4.5
Valuations and the factorization
4.6
Stickelberger’s theorem
5
The Kronecker-Weber theorem
▶
5.1
Reduction to the cyclic case
5.2
Removing the ramification away from \(p\)
▶
5.2.1
Tame ramification
5.2.2
Abhyankar’s lemma and the inertia field
5.3
Comparing cyclic extensions
5.4
The odd case
▶
5.4.1
The Kummer extension
5.4.2
The class group argument
5.4.3
Cyclic extensions of odd prime power degree
5.5
The case \(p = 2\)
6
Auxiliary results
Dependency graph
SKW
Xavier Roblot
1
Setup
1.1
The fields and the primes
1.2
The Teichmüller character
1.3
The additive character
1.4
Primes above \(p\)
2
The Gauss sum
2.1
Gauss and Jacobi sums
2.2
Congruences
2.3
Galois action
3
The valuation function \(s\)
3.1
General properties
3.2
Upper bound by the sum of digits
3.3
The digit formula
4
Ideal factorization of Gauss sums
4.1
Setup for Stickelberger’s theorem
4.2
Descent of \(g_d^m\) to \(k\)
4.3
Factorization in \(L\)
4.4
Primes of \(k\) above \(p\)
4.5
Valuations and the factorization
4.6
Stickelberger’s theorem
5
The Kronecker-Weber theorem
5.1
Reduction to the cyclic case
5.2
Removing the ramification away from \(p\)
5.2.1
Tame ramification
5.2.2
Abhyankar’s lemma and the inertia field
5.3
Comparing cyclic extensions
5.4
The odd case
5.4.1
The Kummer extension
5.4.2
The class group argument
5.4.3
Cyclic extensions of odd prime power degree
5.5
The case \(p = 2\)
6
Auxiliary results