Kummer’s Criterion
1
Introduction
▼
1
Overview
2
Notation and conventions
3
Outline
2
Class numbers of the cyclotomic CM field
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1
The real subfield and the class numbers
2
Descent of principal ideals
3
The relative class number
3
The minus class-number criterion
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1
Characters and Bernoulli factors
2
The relative class-number formula
3
From \(h^-\) to Bernoulli numerators
4
Cyclotomic units and Kummer logarithms
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1
The real cyclotomic-unit subgroup
2
Exact \(p\)-saturation
3
The Kummer logarithm matrix
4
The prime-conductor index theorem
5
Final assembly
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1
The implication from \(h^+\) to \(h^-\)
2
The total class-number criterion
3
Kummer’s criterion
6
Infinitely many irregular primes
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1
From Bernoulli numerators to irregular primes
2
Numerators, \(p\)-adic congruences, and von Staudt–Clausen
3
Growth of divided Bernoulli numbers
4
The divisor-closed multiplier
5
The unrestricted Kummer congruence
6
The Carlitz escape argument
Dependency graph
1 Introduction
1
Overview
2
Notation and conventions
3
Outline