# Cyclotomic field

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{{Short description|Field extension of the rational numbers by a primitive root of unity}}
{{more footnotes needed|date=September 2012}}
In [algebraic number theory](/source/algebraic_number_theory), a '''cyclotomic field''' is a [number field](/source/number_field) obtained by [adjoining](/source/adjunction_(field_theory)) a [complex](/source/complex_number) [root of unity](/source/root_of_unity) to <math>\Q</math>, the [field](/source/field_(mathematics)) of [rational number](/source/rational_number)s.<ref>{{Cite book |url=https://link.springer.com/book/10.1007/978-1-4757-3976-3 |title=Elements of Algebra |series=Undergraduate Texts in Mathematics |date=1994 |publisher=Springer New York |pages=100 |language=en |doi=10.1007/978-1-4757-3976-3|isbn=978-1-4419-2839-9 |last1=Stillwell |first1=John }}</ref>

Cyclotomic fields played a crucial role in the development of modern [algebra](/source/abstract_algebra) and number theory because of their relation with [Fermat's Last Theorem](/source/Fermat's_Last_Theorem). It was in the process of his deep investigations of the arithmetic of these fields (for [prime](/source/prime_number) <math>n</math>)—and more precisely, because of the failure of [unique factorization](/source/unique_factorization) in their [rings of integers](/source/ring_of_integers)—that [Ernst Kummer](/source/Ernst_Kummer) first introduced the concept of an [ideal number](/source/ideal_number) and proved his celebrated [congruences](/source/Kummer's_congruences).

==Definition==

For <math>n \geq 1</math>, let 
:<math>\zeta_n=e^{2\pi i/n}\in\C.</math>
This is a [primitive](/source/primitive_root_of_unity) <math>n</math>th root of unity.  Then the <math>n</math>th cyclotomic field is the [field extension](/source/field_extension) <math>\mathbb{Q}(\zeta_n)</math> of <math>\mathbb{Q}</math> generated by <math>\zeta_n</math>.

==Properties==
* The <math>n</math>th [cyclotomic polynomial](/source/cyclotomic_polynomial) <math display="block">
\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](/source/irreducible_polynomial), so it is the [minimal polynomial](/source/Minimal_polynomial_(field_theory)) of <math display="inline">\zeta_n</math> over <math display="inline">\Q</math>.

* The [conjugates](/source/conjugate_element_(field_theory)) of <math>\zeta_n</math> in <math>\C</math> are therefore the other primitive <math>n</math>-th roots of unity: <math>\zeta_n^k</math> for <math>1\leq k\leq n</math> with <math>\gcd(k,n)=1</math>.

* The [degree](/source/degree_of_a_field_extension) of <math>\Q(\zeta_n)</math> is therefore <math>[\Q(\zeta_n):\Q]=\deg\Phi_n=\varphi(n)</math>, where <math>\varphi</math> is [Euler's totient function](/source/Euler's_totient_function).

* The [roots](/source/root_of_a_polynomial) of <math>x^n-1</math> are the powers of <math>\zeta_n</math>, so <math>\Q(\zeta_n)</math> is the [splitting field](/source/splitting_field) of <math>x^n-1</math> (or of <math>\Phi_n</math>) over <math>\Q</math>. It follows that <math>\Q(\zeta_n)</math> is a [Galois extension](/source/Galois_extension) of <math>\mathbb{Q}</math>.

* The [Galois group](/source/Galois_group) <math>\operatorname{Gal}(\Q(\zeta_n)/\Q)</math> is [naturally isomorphic](/source/natural_transformation) to [the multiplicative group <math>(\Z/n\Z)^\times</math>](/source/multiplicative_group_of_integers_modulo_n), which consists of the invertible residues [modulo](/source/modular_arithmetic) <math>n</math>, which are the residues <math>a</math> mod <math>n</math> with <math>1\leq a \leq n</math> and <math>\gcd(a,n)=1</math>. The [isomorphism](/source/isomorphism) sends each <math>\sigma \in \operatorname{Gal}(\Q(\zeta_n)/\Q)</math> to <math>a</math> mod <math>n</math>, where <math>a</math> is an [integer](/source/integer) such that <math>\sigma(\zeta_n)=\zeta_n^a</math>.

* The [ring of integers](/source/ring_of_integers) of <math>\Q(\zeta_n)</math> is <math>\Z[\zeta_n]</math>.

* For <math>n>2</math>, the [discriminant](/source/Discriminant_of_an_algebraic_number_field) of the extension <math>\Q(\zeta_n)/\Q</math> 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(\zeta_n)/\Q</math> is [unramified](/source/unramified) above every prime not dividing <math>n</math>.

* If <math>n</math> is a power of a prime <math>p</math>, then <math>\Q(\zeta_n)/\Q</math> is totally ramified above <math>p</math>.

* If <math>q</math> is a prime not dividing <math>n</math>, then the [Frobenius element](/source/Frobenius_element) <math>\operatorname{Frob}_q \in \operatorname{Gal}(\Q(\zeta_n)/\Q)</math> corresponds to the residue of <math>q</math> in <math>(\Z/n\Z)^\times</math>.

* The group of roots of unity in <math>\Q(\zeta_n)</math> has order <math>n</math> or <math>2n</math>, according to whether <math>n</math> is even or odd.

* The [unit group](/source/unit_group) <math>\Z[\zeta_n]^\times</math> is a [finitely generated abelian group](/source/finitely_generated_abelian_group) of rank <math>\varphi(n)/2-1</math>, for any <math>n>2</math>, by the [Dirichlet unit theorem](/source/Dirichlet_unit_theorem). In particular, <math>\Z[\zeta_n]^\times</math> is [finite](/source/finite_group) only for <math>n\in\{1,2,3,4,6\}</math>. The [torsion subgroup](/source/torsion_subgroup) of <math>\Z[\zeta_n]^\times</math> is the group of roots of unity in <math>\Q(\zeta_n)</math>, which was described in the previous item.  [Cyclotomic unit](/source/Cyclotomic_unit)s form an explicit finite-[index](/source/index_of_a_subgroup) [subgroup](/source/subgroup) of <math>\Z[\zeta_n]^\times</math>.

* The [Kronecker–Weber theorem](/source/Kronecker%E2%80%93Weber_theorem) states that every [finite](/source/finite_extension) [abelian extension](/source/abelian_extension) of <math>\Q</math> in <math>\C</math> is contained in <math>\Q(\zeta_n)</math> for some <math>n</math>.  Equivalently, the union of all the cyclotomic fields <math>\Q(\zeta_n)</math> is the [maximal abelian extension](/source/maximal_abelian_extension) <math>\Q^{\mathrm{ab}}</math> of <math>\Q</math>.

== Relation with regular polygons ==

[Gauss](/source/Carl_Friedrich_Gauss) made early inroads in the theory of cyclotomic fields, in connection with the problem of [constructing](/source/Constructible_polygon) a [regular {{mvar|n}}-gon](/source/regular_polygon) with a [compass and straightedge](/source/compass_and_straightedge). His surprising result that had escaped his predecessors was that a regular [17-gon](/source/heptadecagon) could be so constructed.  More generally, for any integer <math>n\geq 3</math>, the following are equivalent:
* a regular <math>n</math>-gon is constructible;
* there is a sequence of fields, starting with <math>\Q</math> and ending with <math>\Q(\zeta_n)</math>, such that each is a [quadratic extension](/source/quadratic_extension) of the previous field;
* <math>\varphi(n)</math> is a [power of 2](/source/power_of_2);
* <math>n=2^a p_1 \cdots p_r</math> for some integers <math>a, r\geq 0</math> and [Fermat prime](/source/Fermat_prime)s <math>p_1,\ldots,p_r</math>.  (A Fermat prime is an odd prime <math>p</math> such that <math>p-1</math> is a power of 2.  The known Fermat primes are [3](/source/3_(number)), [5](/source/5_(number)), [17](/source/17_(number)), [257](/source/257_(number)), [65537](/source/65537_(number)), and it is likely that there are no others.)

===Small examples===
* <math>n=3</math> and <math>n=6</math>: The equations <math display="inline">\zeta_3 = \tfrac{1}{2}(-1+\sqrt{-3}\,)</math> and <math display="inline">\zeta_6 = \tfrac{1}{2}( 1+\sqrt{-3} \,)</math> show that <math display="inline">\mathbb{Q}(\zeta_3) = \mathbb{Q}(\zeta_6)=\mathbb{Q}(\sqrt{-3})</math>, which is a quadratic extension of <math display="inline">\mathbb{Q}</math>. Correspondingly, a regular 3-gon and a regular 6-gon are constructible.
* <math>n=4</math>: Similarly, {{math|1=ζ<sub>4</sub> = ''i''}}, so <math display="inline">\mathbb{Q}(\zeta_4)</math>, and a regular 4-gon is constructible.
* <math>n=5</math>: The field <math display="inline">\mathbb{Q}(\zeta_5)</math> is not a quadratic extension of <math display="inline">\mathbb{Q}</math>, but it is a quadratic extension of the quadratic extension <math display="inline">\mathbb{Q}(\sqrt{5})</math>, so a regular 5-gon is constructible.

== Relation with Fermat's Last Theorem ==

A natural approach to proving [Fermat's Last Theorem](/source/Fermat's_Last_Theorem) is to factor the binomial <math>x^n + y^n</math>,
where <math>n</math> 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_n y)\ldots (x + \zeta_n^{n-1} y)</math>

Here <math>x</math> and <math>y</math> are ordinary integers, whereas the factors are [algebraic integer](/source/algebraic_integer)s in the cyclotomic field <math>\Q(\zeta_n)</math>. If [unique factorization](/source/fundamental_theorem_of_arithmetic) holds in the cyclotomic integers <math>\Z[\zeta_n]</math>, 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>n=4</math> and Euler's proof for <math>n=3</math> can be recast in these terms. The complete list of {{mvar|n}} for which <math>\Z[\zeta_n]</math> 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.

[Kummer](/source/Ernst_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[\zeta_n]</math>, measured the failure of unique factorization via the [class number](/source/class_number_(number_theory)) <math>h_n</math> and proved that if <math>h_p</math> is not divisible by a prime <math>p</math> (such <math>p</math> are called ''[regular prime](/source/regular_prime)s'') then Fermat's theorem is true for the exponent <math>n=p</math>. Furthermore, he [gave a criterion](/source/Kummer's_criterion) to determine which primes are regular, and established Fermat's theorem for all prime exponents <math>p</math> less than 100, except for the ''irregular primes'' [37](/source/37_(number)), [59](/source/59_(number)), and [67](/source/67_(number)). Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by [Iwasawa](/source/Kenkichi_Iwasawa) in [Iwasawa theory](/source/Iwasawa_theory) and by Kubota and Leopoldt in their theory of [<math>p</math>-adic zeta functions](/source/p-adic_zeta_function).

== 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](/source/Kronecker%E2%80%93Weber_theorem)
*[Cyclotomic polynomial](/source/Cyclotomic_polynomial)

==References==
{{Reflist}}

===Sources===
* [Bryan Birch](/source/Bryan_John_Birch), "Cyclotomic fields and Kummer extensions", in [J.W.S. Cassels](/source/J.W.S._Cassels) and [A. Frohlich](/source/Albrecht_Fr%C3%B6hlich) (edd), ''Algebraic number theory'', [Academic Press](/source/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](/source/Serge_Lang), ''Cyclotomic Fields I and II'', Combined second edition. With an appendix by [Karl Rubin](/source/Karl_Rubin). [Graduate Texts in Mathematics](/source/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](/source/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}}
* {{cite journal | last=Yamagata | first=Koji | last2=Yamagishi | first2=Masakazu | title=On the ring of integers of real cyclotomic fields | journal=Proc. Japan Academy, Series A, Math. Sciences | volume=92 | issue=6 | date=2016 | issn=0386-2194 | doi=10.3792/pjaa.92.73 | doi-access=free}}

Category:Algebraic number theory
*

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