Cyclotomic field#Cyclotomic fields

For $n\; \backslash geq\; 1$, let

:$\backslash zeta\_n=\backslash left\backslash \{e^\{\backslash frac\{2k\backslash pi\; i\}n\}\backslash ;\; |\; n,k\; \backslash in\; \backslash N\; \backslash ;\; \backslash gcd(n,k)=1\backslash right\backslash \}$.

This is a primitive $n$th root of unity. Then the $n$th cyclotomic field is the field extension $\backslash mathbb\{Q\}(\backslash zeta\_n)$ of $\backslash mathbb\{Q\}$ generated by $\backslash zeta\_n$.

• The $n$th cyclotomic polynomial

::$$

\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)

:is irreducible, so it is the minimal polynomial of $\backslash zeta\_n$ over $\backslash Q$.

• The conjugates of $\backslash zeta\_n$ in $\backslash C$ are therefore the other primitive {{mvar|n}}th roots of unity: $\backslash zeta\_n^k$ for $1\backslash leq\; k\backslash leq\; n$ with $\backslash gcd(k,n)=1$.

• The degree of $\backslash Q(\backslash zeta\_n)$ is therefore $[\backslash Q(\backslash zeta\_n):\backslash Q]=\backslash deg\backslash Phi\_n=\backslash varphi(n)$, where $\backslash varphi$ is Euler's totient function.

• The roots of $x^n-1$ are the powers of $\backslash zeta\_n$, so $\backslash Q(\backslash zeta\_n)$ is the splitting field of $x^n-1$ (or of $\backslash Phi\_n$) over $\backslash Q$. It follows that $\backslash Q(\backslash zeta\_n)$ is a Galois extension of $\backslash mathbb\{Q\}$.

• The Galois group $\backslash operatorname\{Gal\}(\backslash Q(\backslash zeta\_n)/\backslash Q)$ is naturally isomorphic to the multiplicative group $(\backslash Z/n\backslash Z)^\backslash times$, which consists of the invertible residues modulo $n$, which are the residues $a$ mod $n$ with $1\backslash leq\; a\; \backslash leq\; n$ and $\backslash gcd(a,n)=1$. The isomorphism sends each $\backslash sigma\; \backslash in\; \backslash operatorname\{Gal\}(\backslash Q(\backslash zeta\_n)/\backslash Q)$ to $a$ mod $n$, where $a$ is an integer such that $\backslash sigma(\backslash zeta\_n)=\backslash zeta\_n^a$.

• The ring of integers of $\backslash Q(\backslash zeta\_n)$ is $\backslash Z[\backslash zeta\_n]$.

• For $n>2$, the discriminant of the extension $\backslash Q(\backslash zeta\_n)/\backslash Q$ is{{sfn|Washington|1997|loc=Proposition 2.7}}

:: $(-1)^\{\backslash varphi(n)/2\}\backslash ,$

\frac{n^{\varphi(n)}}

{\displaystyle\prod_{p|n} p^{\varphi(n)/(p-1)}}.

• In particular, $\backslash Q(\backslash zeta\_n)/\backslash Q$ is unramified above every prime not dividing $n$.

• If $n$ is a power of a prime $p$, then $\backslash Q(\backslash zeta\_n)/\backslash Q$ is totally ramified above $p$.

• If $q$ is a prime not dividing $n$, then the Frobenius element $\backslash operatorname\{Frob\}\_q\; \backslash in\; \backslash operatorname\{Gal\}(\backslash Q(\backslash zeta\_n)/\backslash Q)$ corresponds to the residue of $q$ in $(\backslash Z/n\backslash Z)^\backslash times$.

• The group of roots of unity in $\backslash Q(\backslash zeta\_n)$ has order $n$ or $2n$, according to whether $n$ is even or odd.

• The unit group $\backslash Z[\backslash zeta\_n]^\backslash times$ is a finitely generated abelian group of rank $\backslash varphi(n)/2-1$, for any $n>2$, by the Dirichlet unit theorem. In particular, $\backslash Z[\backslash zeta\_n]^\backslash times$ is finite only for $n\backslash in\backslash \{1,2,3,4,6\backslash \}$. The torsion subgroup of $\backslash Z[\backslash zeta\_n]^\backslash times$ is the group of roots of unity in $\backslash Q(\backslash zeta\_n)$, which was described in the previous item. Cyclotomic units form an explicit finite-index subgroup of $\backslash Z[\backslash zeta\_n]^\backslash times$.

• The Kronecker–Weber theorem states that every finite abelian extension of $\backslash Q$ in $\backslash C$ is contained in $\backslash Q(\backslash zeta\_n)$ for some $n$. Equivalently, the union of all the cyclotomic fields $\backslash Q(\backslash zeta\_n)$ is the maximal abelian extension $\backslash Q^\{\backslash mathrm\{ab\}\}$ of $\backslash Q$.

Gauss made early inroads in the theory of cyclotomic fields, in connection with the problem of constructing a regular polygon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular 17-gon could be so constructed. More generally, for any integer $n\backslash geq\; 3$, the following are equivalent:

• a regular $n$-gon is constructible;
• there is a sequence of fields, starting with $\backslash Q$ and ending with $\backslash Q(\backslash zeta\_n)$, such that each is a quadratic extension of the previous field;
• $\backslash varphi(n)$ is a power of 2;
• $n=2^a\; p\_1\; \backslash cdots\; p\_r$ for some integers $a,\; r\backslash geq\; 0$ and Fermat primes $p\_1,\backslash ldots,p\_r$. (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, 5, 17, 257, 65537, and it is likely that there are no others.)

{{MOS|section|MOS:RADICAL|date=May 2025}}

• {{math|1=n = 3}} and {{math|1=n = 6}}: The equations $\backslash zeta\_3\; =\; \backslash tfrac\{-1+\backslash sqrt\{-3\}\}\{2\}$ and $\backslash zeta\_6\; =\; \backslash tfrac\{1+\backslash sqrt\{-3\}\}\{2\}$ show that {{math|1=Q(ζ₃) = Q(ζ₆) = Q({{sqrt|−3}} )}}, 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=ζ₄ = i}}, so {{math|1=Q(ζ₄) = Q(i)}}, and a regular 4-gon is constructible.
• {{math|1=n = 5}}: The field {{math|Q(ζ₅)}} is not a quadratic extension of {{math|Q}}, but it is a quadratic extension of the quadratic extension {{math|Q({{sqrt|5}} )}}, so a regular 5-gon is constructible.

A natural approach to proving Fermat's Last Theorem is to factor the binomial {{math|xⁿ + yⁿ}},

where {{mvar|n}} is an odd prime, appearing in one side of Fermat's equation

: $x^n\; +\; y^n\; =\; z^n$

as follows:

: $x^n\; +\; y^n\; =\; (x\; +\; y)(x\; +\; \backslash zeta\_n\; y)\backslash ldots\; (x\; +\; \backslash zeta\_n^\{n-1\}\; y)$

Here {{mvar|x}} and {{mvar|y}} are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field {{math|Q(ζ_n)}}. If unique factorization holds in the cyclotomic integers {{math|Z[ζ_n]}}, 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_n]}} 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 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_n]}}, measured the failure of unique factorization via the class number {{math|h_n}} and proved that if {{math|h_p}} is not divisible by a prime {{mvar|p}} (such {{mvar|p}} are called regular primes) then Fermat's theorem is true for the exponent {{math|1=n = p}}. Furthermore, he 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, 59, and 67. Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by Iwasawa in Iwasawa theory and by Kubota and Leopoldt in their theory of p-adic zeta functions.

{{OEIS|id=A061653}}, or {{oeis|id=A055513}} or {{oeis|id=A000927}} for the $h$-part (for prime n)

• Kronecker–Weber theorem
• Cyclotomic polynomial

{{Reflist}}

• 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. 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}}

• {{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}}
• {{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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