Cyclotomic field

Template:Short description Template:More footnotes needed In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to $\mathbb{Q}$, the field of rational numbers.

Cyclotomic fields played a crucial role in the development of modern 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 Template:Mvar) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences.

Definition

For $n\ge 1$, let Template:Math; this is a primitive Template:Mvarth root of unity. Then the Template:Mvarth cyclotomic field is the extension $\mathbb{Q}({\zeta }_n)$ of $\mathbb{Q}$ generated by Template:Math.

Properties

• The Template:Mvarth cyclotomic polynomial

${\mathrm{\Phi }}_n(x)=\prod _{\stackrel{1\le k\le n}{\gcd (k,n)=1}}(x-e^{2\pi ik/n})=\prod _{\stackrel{1\le k\le n}{\gcd (k,n)=1}}(x-{{\zeta }_n}^k)$

is irreducible, so it is the minimal polynomial of Template:Math over $\mathbb{Q}$.

• The conjugates of Template:Math in Template:Math are therefore the other primitive Template:Mvarth roots of unity: Template:Math for Template:Math with Template:Math.
• The degree of $\mathbb{Q}({\zeta }_n)$ is therefore Template:Math, where Template:Mvar is Euler's totient function.
• The roots of Template:Math are the powers of Template:Math, so Template:Math is the splitting field of Template:Math (or of Template:Math) over Template:Math.
• Therefore Template:Math is a Galois extension of $\mathbb{Q}$.
• The Galois group $\mathrm{Gal}(𝐐({\zeta }_n)/𝐐)$ is naturally isomorphic to the multiplicative group $(𝐙/n𝐙{)}^{\times }$, which consists of the invertible residues modulo Template:Mvar, which are the residues Template:Math with Template:Math and Template:Math. The isomorphism sends each $\sigma \in \mathrm{Gal}(𝐐({\zeta }_n)/𝐐)$ to Template:Math, where Template:Mvar is an integer such that Template:Math.
• The ring of integers of Template:Math is Template:Math.
• For Template:Math, the discriminant of the extension Template:Math isTemplate:Sfn

$(-1{)}^{\phi (n)/2}\frac{n^{\phi (n)}}{\prod _{p|n}{p}^{\phi (n)/(p-1)}}.$

• In particular, Template:Math is unramified above every prime not dividing Template:Mvar.
• If Template:Mvar is a power of a prime Template:Mvar, then Template:Math is totally ramified above Template:Mvar.
• If Template:Mvar is a prime not dividing Template:Mvar, then the Frobenius element ${\mathrm{Frob}}_q\in \mathrm{Gal}(𝐐({\zeta }_n)/𝐐)$ corresponds to the residue of Template:Math in $(𝐙/n𝐙{)}^{\times }$.
• The group of roots of unity in Template:Math has order Template:Mvar or Template:Math, according to whether Template:Mvar is even or odd.
• The unit group Template:Math is a finitely generated abelian group of rank Template:Math, for any Template:Math, by the Dirichlet unit theorem. In particular, Template:Math is finite only for Template:Math}. The torsion subgroup of Template:Math is the group of roots of unity in Template:Math, which was described in the previous item. Cyclotomic units form an explicit finite-index subgroup of Template:Math.
• The Kronecker–Weber theorem states that every finite abelian extension of Template:Math in Template:Math is contained in Template:Math for some Template:Mvar. Equivalently, the union of all the cyclotomic fields Template:Math is the maximal abelian extension Template:Math of Template:Math.

Relation with regular polygons

Gauss made early inroads in the theory of cyclotomic fields, in connection with the problem of constructing a [[regular polygon|regular Template:Mvar-gon]] 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 Template:Math, the following are equivalent:

• a regular Template:Mvar-gon is constructible;
• there is a sequence of fields, starting with Template:Math and ending with Template:Math, such that each is a quadratic extension of the previous field;
• Template:Math is a power of 2;
• $n=2^a{p}_1\cdots p_r$ for some integers Template:Math and Fermat primes $p_1,\dots ,p_r$. (A Fermat prime is an odd prime Template:Mvar such that Template:Math is a power of 2. The known Fermat primes are 3, 5, 17, 257, 65537, and it is likely that there are no others.)

Small examples

• Template:Math and Template:Math: The equations ${\zeta }_3=\frac{-1+\sqrt{-3}}{2}$ and ${\zeta }_6=\frac{1+\sqrt{-3}}{2}$ show that Template:Math, which is a quadratic extension of Template:Math. Correspondingly, a regular 3-gon and a regular 6-gon are constructible.
• Template:Math: Similarly, Template:Math, so Template:Math, and a regular 4-gon is constructible.
• Template:Math: The field Template:Math is not a quadratic extension of Template:Math, but it is a quadratic extension of the quadratic extension Template:Math, 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 Template:Math, where Template:Mvar is an odd prime, appearing in one side of Fermat's equation

$x^n+y^n=z^n$

as follows:

$x^n+y^n=(x+y)(x+\zeta y)\cdots (x+{\zeta }^{n-1}y)$

Here Template:Mvar and Template:Mvar are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Template:Math. If unique factorization holds in the cyclotomic integers Template: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 Template:Math and Euler's proof for Template:Math can be recast in these terms. The complete list of Template:Mvar for which Template:Math has unique factorization isTemplate:Sfn

• 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 Template:Math, measured the failure of unique factorization via the class number Template:Math and proved that if Template:Math is not divisible by a prime Template:Mvar (such Template:Mvar are called regular primes) then Fermat's theorem is true for the exponent Template:Math. Furthermore, he gave a criterion to determine which primes are regular, and established Fermat's theorem for all prime exponents Template:Mvar 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.

List of class numbers of cyclotomic fields

Template:OEIS, or Template:Oeis or Template:Oeis for the $h$-part (for prime n)

See also

• Kronecker–Weber theorem
• Cyclotomic polynomial

References

Sources

• 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
• Template:Citation
• 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. Template:ISBN

Further reading

• Template:Cite book
• Template:Mathworld
• Template:Springer
• On the Ring of Integers of Real Cyclotomic Fields. Koji Yamagata and Masakazu Yamagishi: Proc, Japan Academy, 92. Ser a (2016)