Cyclotomic field

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In

• The nth cyclotomic polynomial

${\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!\left(x-e^{2\pi ik/n}\right)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!(x-{\zeta _{n}}^{k})}$

is irreducible, so it is the minimal polynomial of ζ_n over Q.

• The conjugates of ζ_n in C are therefore the other primitive nth roots of unity: ζ^k
  _n for 1 ≤ k ≤ n with gcd(k, n) = 1.
• The degree of Q(ζ_n) is therefore [Q(ζ_n) : Q] = deg Φ_n = φ(n), where φ is Euler's totient function.
• The
  roots of xⁿ − 1 are the powers of ζ_n, so Q(ζ_n) is the splitting field
  of xⁿ − 1 (or of Φ(x)) over Q.
• Therefore Q(ζ_n) is a Galois extension of Q.
• The Galois group ${\displaystyle \operatorname {Gal} (\mathbf {Q} (\zeta _{n})/\mathbf {Q} )}$ is naturally isomorphic to the multiplicative group ${\displaystyle (\mathbf {Z} /n\mathbf {Z} )^{\times }}$, which consists of the invertible residues modulo n, which are the residues a mod n with 1 ≤ a ≤ n and gcd(a, n) = 1. The isomorphism sends each ${\displaystyle \sigma \in \operatorname {Gal} (\mathbf {Q} (\zeta _{n})/\mathbf {Q} )}$ to a mod n, where a is an integer such that σ(ζ_n) = ζ^a
  _n.
• The ring of integers of Q(ζ_n) is Z[ζ_n].
• For n > 2, the discriminant of the extension Q(ζ_n) / Q is^[1]

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

• In particular, Q(ζ_n) / Q is
  unramified
  above every prime not dividing n.
• If n is a power of a prime p, then Q(ζ_n) / Q is totally ramified above p.
• If q is a prime not dividing n, then the
  Frobenius element
  ${\displaystyle \operatorname {Frob} _{q}\in \operatorname {Gal} (\mathbf {Q} (\zeta _{n})/\mathbf {Q} )}$ corresponds to the residue of q in ${\displaystyle (\mathbf {Z} /n\mathbf {Z} )^{\times }}$.
• The group of roots of unity in Q(ζ_n) has order n or 2n, according to whether n is even or odd.
• The
  Dirichlet unit theorem. In particular, Z[ζ_n]^× is finite only for n ∈ {1, 2, 3, 4, 6}. The torsion subgroup of Z[ζ_n]^× is the group of roots of unity in Q(ζ_n), which was described in the previous item. Cyclotomic units form an explicit finite-index subgroup
  of Z[ζ_n]^×.
• The
  maximal abelian extension
  Q^ab of Q.

A natural approach to proving Fermat's Last Theorem is to factor the binomial xⁿ + yⁿ, where n is an odd prime, appearing in one side of Fermat's equation

${\displaystyle x^{n}+y^{n}=z^{n}}$

as follows:

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

Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζ_n). If unique factorization holds in the cyclotomic integers 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 n = 4 and Euler's proof for n = 3 can be recast in these terms. The complete list of n for which Z[ζ_n] has unique factorization is^[2]

• 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.

List of class numbers of cyclotomic fields

(sequence A061653 in the OEIS), or OEIS: A055513 or OEIS: A000927 for the ${\displaystyle h}$-part (for prime n)

See also

• Kronecker–Weber theorem
• Cyclotomic polynomial

References

Sources

• A. Frohlich (edd), Algebraic number theory, Academic Press
  , 1973. Chap.III, pp. 45–93.
• Daniel A. Marcus, Number Fields, first edition, Springer-Verlag, 1977
• Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 ed.), New York: Springer-Verlag,
  MR 1421575
• ISBN 0-387-96671-4

• Zbl 1100.11002
  .
• Weisstein, Eric W. "Cyclotomic Field". MathWorld.
• "Cyclotomic field", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• On the Ring of Integers of Real Cyclotomic Fields. Koji Yamagata and Masakazu Yamagishi: Proc, Japan Academy, 92. Ser a (2016)