View source for Cyclotomic field

{{Short description|Field extension of the rational numbers by a primitive root of unity}}
{{more footnotes needed|date=September 2012}}
In [[number theory]], a '''cyclotomic field''' is a [[number field]] obtained by [[adjunction (field theory)|adjoining]] a [[complex number|complex]] [[root of unity]] to <math>\mathbb{Q}</math>, the [[field (mathematics)|field]] of [[rational number]]s<ref>{{Cite book |url=https://link.springer.com/book/10.1007/978-1-4757-3976-3 |title=Elements of Algebra |publisher=Springer New York |pages=100 |language=en |doi=10.1007/978-1-4757-3976-3}}</ref>.

Cyclotomic fields played a crucial role in the development of modern [[abstract algebra|algebra]] and number theory because of their relation with [[Fermat's Last Theorem]]. It was in the process of his deep investigations of the arithmetic of these fields (for [[prime number|prime]]&nbsp;{{mvar|n}})&nbsp;– and more precisely, because of the failure of [[unique factorization]] in their [[ring of integers|rings of integers]]&nbsp;– that [[Ernst Kummer]] first introduced the concept of an [[ideal number]] and proved his celebrated [[Kummer's congruences|congruences]].

==Definition==

For <math>n \ge 1</math>, let {{math|1=ζ<sub>''n''</sub> = ''e''<sup>2π''i''/''n''</sup> ∈ '''C'''}}; this is a [[primitive root of unity|primitive]] {{mvar|n}}th root of unity.  Then the {{mvar|n}}th cyclotomic field is the [[field extension|extension]] <math>\mathbb{Q}(\zeta_n)</math> of <math>\mathbb{Q}</math> generated by {{math|ζ<sub>''n''</sub>}}.

==Properties==
* The {{mvar|n}}th [[cyclotomic polynomial]] 
:<math>
\Phi_n(x) =
\prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}\!\!\!
\left(x-e^{2\pi i k/n}\right)
=
\prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}\!\!\!
(x-{\zeta_n}^k)
</math>
:is [[irreducible polynomial|irreducible]], so it is the [[Minimal polynomial (field theory)|minimal polynomial]] of {{math|ζ<sub>''n''</sub>}} over <math>\mathbb{Q}</math>.
* The [[conjugate element (field theory)|conjugates]] of {{math|ζ<sub>''n''</sub>}} in {{math|'''C'''}} are therefore the other primitive {{mvar|n}}th roots of unity: {{math|ζ{{su|b=''n''|p=''k''}}}} for {{math|1 ≤ ''k'' ≤ ''n''}} with {{math|1=gcd(''k'',&thinsp;''n'')&nbsp;=&thinsp;1}}.
* The [[degree of a field extension|degree]] of <math>\mathbb{Q}(\zeta_n)</math> is therefore {{math|1=['''Q'''(ζ<sub>''n''</sub>) : '''Q'''] = deg&thinsp;Φ<sub>''n''</sub> = ''φ''(''n'')}}, where {{mvar|φ}} is [[Euler's totient function]].
* The [[root of a polynomial|roots]] of {{math|''x''<sup>''n''</sup> − 1}} are the powers of {{math|ζ<sub>''n''</sub>}}, so {{math|'''Q'''(ζ<sub>''n''</sub>)}} is the [[splitting field]] of {{math|''x''<sup>''n''</sup> − 1}} (or of {{math|Φ(''x'')}}) over {{math|'''Q'''}}.
* Therefore {{math|'''Q'''(ζ<sub>''n''</sub>)}} is a [[Galois extension]] of <math>\mathbb{Q}</math>.
* The [[Galois group]] <math>\operatorname{Gal}(\mathbf{Q}(\zeta_n)/\mathbf{Q})</math> is [[natural transformation|naturally isomorphic]] to [[multiplicative group of integers modulo n|the multiplicative group <math>(\mathbf{Z}/n\mathbf{Z})^\times</math>]], which consists of the invertible residues [[modular arithmetic|modulo]]&nbsp;{{mvar|n}}, which are the residues {{math|''a'' mod ''n''}} with {{math|1 ≤ ''a'' ≤ ''n''}} and {{math|1=gcd(''a'',&thinsp;''n'')&nbsp;=&thinsp;1}}.  The [[isomorphism]] sends each <math>\sigma \in \operatorname{Gal}(\mathbf{Q}(\zeta_n)/\mathbf{Q})</math> to {{math|''a'' mod ''n''}}, where {{mvar|a}} is an [[integer]] such that {{math|1=σ(ζ<sub>''n''</sub>) = ζ{{su|b=''n''|p=''a''}}}}.
* The [[ring of integers]] of {{math|'''Q'''(ζ<sub>''n''</sub>)}} is {{math|'''Z'''[ζ<sub>''n''</sub>]}}.
* For {{math|''n'' > 2}}, the [[Discriminant of an algebraic number field|discriminant]] of the extension {{math|'''Q'''(ζ<sub>''n''</sub>)&thinsp;/&thinsp;'''Q'''}} is{{sfn|Washington|1997|loc=Proposition 2.7}}
:: <math>(-1)^{\varphi(n)/2}\, 
\frac{n^{\varphi(n)}}
{\displaystyle\prod_{p|n} p^{\varphi(n)/(p-1)}}.</math>
* In particular, {{math|'''Q'''(ζ<sub>''n''</sub>)&thinsp;/&thinsp;'''Q'''}} is [[unramified]] above every prime not dividing {{mvar|n}}.
* If {{mvar|n}} is a power of a prime {{mvar|p}}, then {{math|'''Q'''(ζ<sub>''n''</sub>)&thinsp;/&thinsp;'''Q'''}} is totally ramified above {{mvar|p}}.
* If {{mvar|q}} is a prime not dividing {{mvar|n}}, then the [[Frobenius element]] <math>\operatorname{Frob}_q \in \operatorname{Gal}(\mathbf{Q}(\zeta_n)/\mathbf{Q})</math> corresponds to the residue of {{math|''q''}} in <math>(\mathbf{Z}/n\mathbf{Z})^\times</math>.
* The group of roots of unity in {{math|'''Q'''(ζ<sub>''n''</sub>)}} has order {{mvar|n}} or {{math|2''n''}}, according to whether {{mvar|n}} is even or odd.
* The [[unit group]] {{math|'''Z'''[ζ<sub>''n''</sub>]<sup>×</sup>}} is a [[finitely generated abelian group]] of rank {{math|''φ''(''n'')/2 – 1}}, for any {{math|''n'' > 2}}, by the [[Dirichlet unit theorem]].  In particular, {{math|'''Z'''[ζ<sub>''n''</sub>]<sup>×</sup>}} is [[finite group|finite]] only for {{math|''n'' ∈ {1, 2, 3, 4, 6}}}.  The [[torsion subgroup]] of {{math|'''Z'''[ζ<sub>''n''</sub>]<sup>×</sup>}} is the group of roots of unity in {{math|'''Q'''(ζ<sub>''n''</sub>)}}, which was described in the previous item.  [[Cyclotomic unit]]s form an explicit finite-[[index of a subgroup|index]] [[subgroup]] of {{math|'''Z'''[ζ<sub>''n''</sub>]<sup>×</sup>}}.
* The [[Kronecker–Weber theorem]] states that every [[finite extension|finite]] [[abelian extension]] of {{math|'''Q'''}} in {{math|'''C'''}} is contained in {{math|'''Q'''(ζ<sub>''n''</sub>)}} for some {{mvar|n}}.  Equivalently, the union of all the cyclotomic fields {{math|'''Q'''(ζ<sub>''n''</sub>)}} is the [[maximal abelian extension]] {{math|'''Q'''<sup>ab</sup>}} of {{math|'''Q'''}}.

== Relation with regular polygons ==

[[Carl Friedrich Gauss|Gauss]] made early inroads in the theory of cyclotomic fields, in connection with the problem of [[Constructible polygon|constructing]] a [[regular polygon|regular {{mvar|n}}-gon]] with a [[compass and straightedge]]. His surprising result that had escaped his predecessors was that a regular [[heptadecagon|17-gon]] could be so constructed.  More generally, for any integer {{math|''n'' ≥ 3}}, the following are equivalent:
* a regular {{mvar|n}}-gon is constructible;
* there is a sequence of fields, starting with {{math|'''Q'''}} and ending with {{math|'''Q'''(ζ<sub>''n''</sub>)}}, such that each is a [[quadratic extension]] of the previous field;
* {{math|''φ''(''n'')}} is a [[power of 2]];
* <math>n=2^a p_1 \cdots p_r</math> for some integers {{math|''a'', ''r'' ≥ 0}} and [[Fermat prime]]s <math>p_1,\ldots,p_r</math>.  (A Fermat prime is an odd prime {{mvar|p}} such that {{math|''p'' − 1}} is a power of 2.  The known Fermat primes are [[3 (number)|3]], [[5 (number)|5]], [[17 (number)|17]], [[257 (number)|257]], [[65537 (number)|65537]], and it is likely that there are no others.)

===Small examples===
* {{math|1=''n'' = 3}} and {{math|1=''n'' = 6}}: The equations <math>\zeta_3 = \tfrac{-1+\sqrt{-3}}{2}</math> and <math>\zeta_6 = \tfrac{1+\sqrt{-3}}{2}</math> show that {{math|1='''Q'''(ζ<sub>3</sub>) = '''Q'''(ζ<sub>6</sub>) = '''Q'''({{sqrt|−3}}&hairsp;)}}, which is a quadratic extension of {{math|'''Q'''}}.  Correspondingly, a regular 3-gon and a regular 6-gon are constructible.
* {{math|1=''n'' = 4}}: Similarly, {{math|1=ζ<sub>4</sub> = ''i''}}, so {{math|1='''Q'''(ζ<sub>4</sub>) = '''Q'''(''i'')}}, and a regular 4-gon is constructible.
* {{math|1=''n'' = 5}}: The field {{math|'''Q'''(ζ<sub>5</sub>)}} is not a quadratic extension of {{math|'''Q'''}}, but it is a quadratic extension of the quadratic extension {{math|'''Q'''({{sqrt|5}}&hairsp;)}}, so a regular 5-gon is constructible.

== Relation with Fermat's Last Theorem ==

A natural approach to proving [[Fermat's Last Theorem]] is to factor the binomial {{math|''x<sup>n</sup>'' + ''y<sup>n</sup>''}},
where {{mvar|n}} is an odd prime, appearing in one side of Fermat's equation

: <math>x^n + y^n = z^n</math>

as follows:

: <math>x^n + y^n = (x + y)(x + \zeta y)\cdots (x + \zeta^{n-1} y)</math>

Here {{mvar|x}} and {{mvar|y}} are ordinary integers, whereas the factors are [[algebraic integer]]s in the cyclotomic field {{math|'''Q'''(''ζ''<sub>''n''</sub>)}}. If [[fundamental theorem of arithmetic|unique factorization]] holds in the cyclotomic integers {{math|'''Z'''[''ζ''<sub>''n''</sub>]}}, then it can be used to rule out the existence of nontrivial solutions to Fermat's equation.

Several attempts to tackle Fermat's Last Theorem proceeded along these lines, and both Fermat's proof for {{math|1=''n'' = 4}} and Euler's proof for {{math|1=''n'' = 3}} can be recast in these terms. The complete list of {{mvar|n}} for which {{math|'''Z'''[''ζ''<sub>''n''</sub>]}} has unique factorization is{{sfn|Washington|1997|loc=Theorem 11.1}}

* 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90.

[[Ernst Kummer|Kummer]] found a way to deal with the failure of unique factorization. He introduced a replacement for the prime numbers in the cyclotomic integers {{math|'''Z'''[''ζ''<sub>''n''</sub>]}}, measured the failure of unique factorization via the [[class number (number theory)|class number]] {{math|''h<sub>n</sub>''}} and proved that if {{math|''h<sub>p</sub>''}} is not divisible by a prime {{mvar|p}} (such {{mvar|p}} are called ''[[regular prime]]s'') then Fermat's theorem is true for the exponent {{math|1=''n'' = ''p''}}. Furthermore, he [[Kummer's criterion|gave a criterion]] to determine which primes are regular, and established Fermat's theorem for all prime exponents {{mvar|p}} less than 100, except for the ''irregular primes'' [[37 (number)|37]], [[59 (number)|59]], and [[67 (number)|67]]. Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by [[Kenkichi Iwasawa|Iwasawa]] in [[Iwasawa theory]] and by Kubota and Leopoldt in their theory of [[p-adic zeta function|''p''-adic zeta functions]].

== List of class numbers of cyclotomic fields ==

{{OEIS|id=A061653}}, or {{oeis|id=A055513}} or {{oeis|id=A000927}} for the <math>h</math>-part (for prime ''n'')

<div style="overflow:auto">
{{columns-list|colwidth=17em|
* 1-22: 1
* 23: 3
* 24-28: 1
* 29: 8
* 30: 1
* 31: 9
* 32-36: 1
* 37: 37
* 38: 1
* 39: 2
* 40: 1
* 41: 121
* 42: 1
* 43: 211
* 44: 1
* 45: 1
* 46: 3
* 47: 695
* 48: 1
* 49: 43
* 50: 1
* 51: 5
* 52: 3
* 53: 4889
* 54: 1
* 55: 10
* 56: 2
* 57: 9
* 58: 8
* 59: 41241
* 60: 1
* 61: 76301
* 62: 9
* 63: 7
* 64: 17
* 65: 64
* 66: 1
* 67: 853513
* 68: 8
* 69: 69
* 70: 1
* 71: 3882809
* 72: 3
* 73: 11957417
* 74: 37
* 75: 11
* 76: 19
* 77: 1280
* 78: 2
* 79: 100146415
* 80: 5
* 81: 2593
* 82: 121
* 83: 838216959
* 84: 1
* 85: 6205
* 86: 211
* 87: 1536
* 88: 55
* 89: 13379363737
* 90: 1
* 91: 53872
* 92: 201
* 93: 6795
* 94: 695
* 95: 107692
* 96: 9
* 97: 411322824001
* 98: 43
* 99: 2883
* 100: 55
* 101: 3547404378125
* 102: 5
* 103: 9069094643165
* 104: 351
* 105: 13
* 106: 4889
* 107: 63434933542623
* 108: 19
* 109: 161784800122409
* 110: 10
* 111: 480852
* 112: 468
* 113: 1612072001362952
* 114: 9
* 115: 44697909
* 116: 10752
* 117: 132678
* 118: 41241
* 119: 1238459625
* 120: 4
* 121: 12188792628211
* 122: 76301
* 123: 8425472
* 124: 45756
* 125: 57708445601
* 126: 7
* 127: 2604529186263992195
* 128: 359057
* 129: 37821539
* 130: 64
* 131: 28496379729272136525
* 132: 11
* 133: 157577452812
* 134: 853513
* 135: 75961
* 136: 111744
* 137: 646901570175200968153
* 138: 69
* 139: 1753848916484925681747
* 140: 39
* 141: 1257700495
* 142: 3882809
* 143: 36027143124175
* 144: 507
* 145: 1467250393088
* 146: 11957417
* 147: 5874617
* 148: 4827501
* 149: 687887859687174720123201
* 150: 11
* 151: 2333546653547742584439257
* 152: 1666737
* 153: 2416282880
* 154: 1280
* 155: 84473643916800
* 156: 156
* 157: 56234327700401832767069245
* 158: 100146415
* 159: 223233182255
* 160: 31365
}}
</div>

==See also==
*[[Kronecker–Weber theorem]]
*[[Cyclotomic polynomial]]

==References==
{{Reflist}}

===Sources===
* [[Bryan John Birch|Bryan Birch]], "Cyclotomic fields and Kummer extensions", in [[J.W.S. Cassels]] and [[A. Frohlich]] (edd), ''Algebraic number theory'', [[Academic Press]], 1973.  Chap.III, pp.&nbsp;45–93.
* Daniel A. Marcus, ''Number Fields'', first edition, Springer-Verlag, 1977
* {{citation|first=Lawrence C.|last= Washington|title=Introduction to Cyclotomic Fields|series=Graduate Texts in Mathematics|volume= 83|publisher=Springer-Verlag|place= New York|year= 1997|edition=2|isbn=0-387-94762-0 |mr=1421575|doi=10.1007/978-1-4612-1934-7}}
* [[Serge Lang]], ''Cyclotomic Fields I and II'', Combined second edition. With an appendix by [[Karl Rubin]]. [[Graduate Texts in Mathematics]], 121. Springer-Verlag, New York, 1990. {{ISBN|0-387-96671-4}}

==Further reading==
* {{cite book | first1=John | last1=Coates | authorlink1=John Coates (mathematician) | first2=R. | last2=Sujatha | authorlink2=Sujatha Ramdorai | title=Cyclotomic Fields and Zeta Values | series=Springer Monographs in Mathematics | publisher=[[Springer-Verlag]] | year=2006 | isbn=3-540-33068-2 | zbl=1100.11002 }}
* {{mathworld|urlname=CyclotomicField|title=Cyclotomic Field}}
* {{springer|title=Cyclotomic field|id=p/c027570}}
* On the Ring of Integers of Real Cyclotomic Fields. Koji Yamagata and Masakazu Yamagishi: Proc, Japan Academy, 92. Ser a (2016)

[[Category:Algebraic number theory]]
[[Category:Cyclotomic fields|*]]