Algebraic
Number Theory
J.S. Milne
Version 3.08
July 19, 2020
, An algebraic number field is a finite extension of Q; an algebraic number is an element
of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic
number fields — the ring of integers in the number field, the ideals and units in the ring of
integers, the extent to which unique factorization holds, and so on.
An abelian extension of a field is a Galois extension of the field with abelian Galois
group. Class field theory describes the abelian extensions of a number field in terms of the
arithmetic of the field.
These notes are concerned with algebraic number theory, and the sequel with class field
theory.
BibTeX information
@misc{milneANT,
author={Milne, James S.},
title={Algebraic Number Theory (v3.08)},
year={2020},
note={Available at www.jmilne.org/math/},
pages={166}
}
v2.01 (August 14, 1996). First version on the web.
v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages.
v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages.
v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages.
v3.02 (April 30, 2009). Minor fixes; changed chapter and page styles; 164 pages.
v3.03 (May 29, 2011). Minor fixes; 167 pages.
v3.04 (April 12, 2012). Minor fixes.
v3.05 (March 21, 2013). Minor fixes.
v3.06 (May 28, 2014). Minor fixes; 164 pages.
v3.07 (March 18, 2017). Minor fixes; 165 pages.
v3.08 (July 19, 2020). Minor fixes; 166 pages.
Available at www.jmilne.org/math/
Please send comments and corrections to me at jmilne at umich dot edu.
The photograph is of the Fork Hut, Huxley Valley, New Zealand.
Copyright c 1996–2020 J.S. Milne.
Single paper copies for noncommercial personal use may be made without explicit permission
from the copyright holder.
,Contents
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1 Preliminaries from Commutative Algebra 14
Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Ideals in products of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Noetherian modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Rings of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
The Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Review of tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Rings of Integers 25
First proof that the integral elements form a ring . . . . . . . . . . . . . . . . . . 25
Dedekind’s proof that the integral elements form a ring . . . . . . . . . . . . . . 26
Integral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Review of bases of A-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Review of norms and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Review of bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Rings of integers are finitely generated . . . . . . . . . . . . . . . . . . . . . . . 35
Finding the ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Algorithms for finding the ring of integers . . . . . . . . . . . . . . . . . . . . . 40
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Dedekind Domains; Factorization 46
Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Unique factorization of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
The ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Integral closures of Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . 57
Modules over Dedekind domains (sketch). . . . . . . . . . . . . . . . . . . . . . 57
Factorization in extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
The primes that ramify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1
, Finding factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Examples of factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Eisenstein extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 The Finiteness of the Class Number 68
Norms of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Statement of the main theorem and its consequences . . . . . . . . . . . . . . . . 70
Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Some calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Finiteness of the class number . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Binary quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 The Unit Theorem 85
Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Proof that UK is finitely generated . . . . . . . . . . . . . . . . . . . . . . . . . 87
Computation of the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
S -units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Example: CM fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Example: real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Example: cubic fields with negative discriminant . . . . . . . . . . . . . . . . . 92
Finding .K/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Finding a system of fundamental units . . . . . . . . . . . . . . . . . . . . . . . 93
Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Cyclotomic Extensions; Fermat’s Last Theorem. 95
The basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Class numbers of cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . 101
Units in cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
The first case of Fermat’s last theorem for regular primes . . . . . . . . . . . . . 102
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 Absolute Values; Local Fields 105
Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Nonarchimedean absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Equivalent absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Complete list of absolute values for the rational numbers . . . . . . . . . . . . . 110
The primes of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
The weak approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 113
Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Completions in the nonarchimedean case . . . . . . . . . . . . . . . . . . . . . . 116
Newton’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Extensions of nonarchimedean absolute values . . . . . . . . . . . . . . . . . . . 123
Newton’s polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Locally compact fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Number Theory
J.S. Milne
Version 3.08
July 19, 2020
, An algebraic number field is a finite extension of Q; an algebraic number is an element
of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic
number fields — the ring of integers in the number field, the ideals and units in the ring of
integers, the extent to which unique factorization holds, and so on.
An abelian extension of a field is a Galois extension of the field with abelian Galois
group. Class field theory describes the abelian extensions of a number field in terms of the
arithmetic of the field.
These notes are concerned with algebraic number theory, and the sequel with class field
theory.
BibTeX information
@misc{milneANT,
author={Milne, James S.},
title={Algebraic Number Theory (v3.08)},
year={2020},
note={Available at www.jmilne.org/math/},
pages={166}
}
v2.01 (August 14, 1996). First version on the web.
v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages.
v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages.
v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages.
v3.02 (April 30, 2009). Minor fixes; changed chapter and page styles; 164 pages.
v3.03 (May 29, 2011). Minor fixes; 167 pages.
v3.04 (April 12, 2012). Minor fixes.
v3.05 (March 21, 2013). Minor fixes.
v3.06 (May 28, 2014). Minor fixes; 164 pages.
v3.07 (March 18, 2017). Minor fixes; 165 pages.
v3.08 (July 19, 2020). Minor fixes; 166 pages.
Available at www.jmilne.org/math/
Please send comments and corrections to me at jmilne at umich dot edu.
The photograph is of the Fork Hut, Huxley Valley, New Zealand.
Copyright c 1996–2020 J.S. Milne.
Single paper copies for noncommercial personal use may be made without explicit permission
from the copyright holder.
,Contents
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1 Preliminaries from Commutative Algebra 14
Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Ideals in products of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Noetherian modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Rings of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
The Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Review of tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Rings of Integers 25
First proof that the integral elements form a ring . . . . . . . . . . . . . . . . . . 25
Dedekind’s proof that the integral elements form a ring . . . . . . . . . . . . . . 26
Integral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Review of bases of A-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Review of norms and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Review of bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Rings of integers are finitely generated . . . . . . . . . . . . . . . . . . . . . . . 35
Finding the ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Algorithms for finding the ring of integers . . . . . . . . . . . . . . . . . . . . . 40
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Dedekind Domains; Factorization 46
Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Unique factorization of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
The ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Integral closures of Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . 57
Modules over Dedekind domains (sketch). . . . . . . . . . . . . . . . . . . . . . 57
Factorization in extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
The primes that ramify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1
, Finding factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Examples of factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Eisenstein extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 The Finiteness of the Class Number 68
Norms of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Statement of the main theorem and its consequences . . . . . . . . . . . . . . . . 70
Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Some calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Finiteness of the class number . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Binary quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 The Unit Theorem 85
Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Proof that UK is finitely generated . . . . . . . . . . . . . . . . . . . . . . . . . 87
Computation of the rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
S -units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Example: CM fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Example: real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Example: cubic fields with negative discriminant . . . . . . . . . . . . . . . . . 92
Finding .K/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Finding a system of fundamental units . . . . . . . . . . . . . . . . . . . . . . . 93
Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Cyclotomic Extensions; Fermat’s Last Theorem. 95
The basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Class numbers of cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . 101
Units in cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
The first case of Fermat’s last theorem for regular primes . . . . . . . . . . . . . 102
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 Absolute Values; Local Fields 105
Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Nonarchimedean absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Equivalent absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Complete list of absolute values for the rational numbers . . . . . . . . . . . . . 110
The primes of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
The weak approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 113
Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Completions in the nonarchimedean case . . . . . . . . . . . . . . . . . . . . . . 116
Newton’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Extensions of nonarchimedean absolute values . . . . . . . . . . . . . . . . . . . 123
Newton’s polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Locally compact fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
