Cyclotomic Fields

Roots of Unity

One of the most important classes of number fields arises from the solutions of the equation

x^n=1.

The solutions are called the $n$ -th roots of unity.

In the complex numbers, these roots are

1,\zeta_n,\zeta_n^2,\ldots,\zeta_n^{n-1},

where

\zeta_n=e^{2\pi i/n}.

Geometrically, these numbers lie equally spaced around the unit circle in the complex plane.

For example, the fourth roots of unity are

1,\quad i,\quad -1,\quad -i.

The sixth roots of unity form a regular hexagon.

Roots of unity connect algebra, geometry, trigonometry, and number theory. The fields generated by them are called cyclotomic fields.

Cyclotomic Polynomials

The polynomial

x^n-1

has all $n$ -th roots of unity as roots. However, many of these roots also satisfy equations of smaller degree. The primitive roots play the essential role.

Definition. An $n$ -th root of unity $\zeta$ is called primitive if

\zeta^n=1

but

\zeta^k\neq1 \qquad \text{for } 1\le k<n.

Equivalently, $\zeta$ generates all $n$ -th roots of unity through its powers.

The primitive $n$ -th roots are precisely

\zeta_n^a

with

\gcd(a,n)=1.

The polynomial whose roots are the primitive $n$ -th roots of unity is called the $n$ -th cyclotomic polynomial.

Definition.

\Phi_n(x) = \prod_{\substack{1\le a\le n \\ \gcd(a,n)=1}} (x-\zeta_n^a).

The degree of $\Phi_n(x)$ equals Euler’s totient function:

\deg \Phi_n(x)=\varphi(n).

For example,

\Phi_1(x)=x-1,

\Phi_2(x)=x+1,

\Phi_3(x)=x^2+x+1,

and

\Phi_4(x)=x^2+1.

The factorization

x^n-1 = \prod_{d\mid n}\Phi_d(x)

is fundamental in algebra and number theory.

Definition of Cyclotomic Fields

The field generated by a primitive $n$ -th root of unity is called a cyclotomic field.

Definition.

\mathbb{Q}(\zeta_n)

is the $n$ -th cyclotomic field.

Since all primitive $n$ -th roots are powers of one another, adjoining a single primitive root automatically adjoins all of them.

For example,

\mathbb{Q}(\zeta_3) = \mathbb{Q}\left( \frac{-1+\sqrt{-3}}{2} \right).

Because

\zeta_3^2+\zeta_3+1=0,

the field has degree

[\mathbb{Q}(\zeta_3):\mathbb{Q}]=2.

Similarly,

\mathbb{Q}(\zeta_4)=\mathbb{Q}(i).

Cyclotomic fields are finite extensions of $\mathbb{Q}$ , hence number fields.

Irreducibility of Cyclotomic Polynomials

Cyclotomic polynomials are irreducible over $\mathbb{Q}$ .

Theorem. For every positive integer $n$ ,

\Phi_n(x)

is irreducible in $\mathbb{Q}[x]$ .

Consequently, the minimal polynomial of $\zeta_n$ over $\mathbb{Q}$ is exactly $\Phi_n(x)$ . Therefore,

[\mathbb{Q}(\zeta_n):\mathbb{Q}] = \varphi(n).

This result gives an explicit infinite family of algebraic extensions with computable degree.

For example,

\Phi_5(x) = x^4+x^3+x^2+x+1,

[\mathbb{Q}(\zeta_5):\mathbb{Q}]=4.

Galois Groups of Cyclotomic Fields

Cyclotomic fields are Galois extensions of $\mathbb{Q}$ .

Every automorphism is determined by where it sends $\zeta_n$ . Since automorphisms preserve algebraic relations, $\zeta_n$ must map to another primitive $n$ -th root:

\sigma(\zeta_n)=\zeta_n^a, \qquad \gcd(a,n)=1.

Thus the Galois group is naturally identified with the multiplicative group of units modulo $n$ :

\mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\times.

The group operation corresponds to exponent multiplication modulo $n$ .

For example,

(\mathbb{Z}/5\mathbb{Z})^\times = \{1,2,3,4\},

which is cyclic of order $4$ . Therefore

\mathrm{Gal}(\mathbb{Q}(\zeta_5)/\mathbb{Q}) \cong C_4.

Cyclotomic fields thus provide explicit examples where Galois groups can be described concretely.

Cyclotomic Extensions and Abelian Groups

Since

(\mathbb{Z}/n\mathbb{Z})^\times

is always abelian, cyclotomic extensions are abelian Galois extensions.

This fact lies at the heart of class field theory.

Kronecker and Weber proved a remarkable theorem:

Kronecker-Weber Theorem. Every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field.

Thus cyclotomic fields encode all abelian extensions of the rational numbers.

This theorem reveals that roots of unity govern much of algebraic number theory.

Gaussian Integers and Cyclotomic Fields

The simplest nontrivial cyclotomic field is

\mathbb{Q}(i)=\mathbb{Q}(\zeta_4).

Its ring of integers is

\mathbb{Z}[i] = \{a+bi : a,b\in\mathbb{Z}\},

called the Gaussian integers.

Similarly,

\mathbb{Q}(\zeta_3) = \mathbb{Q}(\sqrt{-3})

has ring of integers

\mathbb{Z}[\zeta_3].

These rings possess rich arithmetic structures involving primes, units, and factorization.

For example, primes congruent to $1\pmod4$ factor in $\mathbb{Z}[i]$ , while primes congruent to $3\pmod4$ remain prime.

Thus congruence conditions on ordinary integers become geometric statements inside cyclotomic fields.

Cyclotomic Fields and Fermat’s Last Theorem

Cyclotomic fields played a major historical role in attempts to prove Fermat’s Last Theorem.

Kummer studied the arithmetic of

\mathbb{Q}(\zeta_p)

for prime $p$ . He discovered that ordinary unique factorization can fail in these fields. To repair this failure, he introduced ideal numbers, which later evolved into Dedekind ideals.

Kummer proved Fermat’s Last Theorem for a large class of primes called regular primes by studying the arithmetic of cyclotomic fields.

Thus cyclotomic fields directly motivated the development of algebraic number theory.

Cyclotomic Fields in Modern Mathematics

Cyclotomic fields appear throughout modern mathematics.

They arise in:

reciprocity laws;
class field theory;
modular forms;
Galois representations;
algebraic topology;
Fourier analysis;
cryptography.

The discrete Fourier transform is fundamentally built from roots of unity. Finite field analogues of cyclotomic constructions appear in coding theory and cryptographic algorithms.

In modern arithmetic geometry, cyclotomic characters describe how Galois groups act on roots of unity, linking field extensions with deep arithmetic invariants.

Thus cyclotomic fields form one of the central bridges between algebra, arithmetic, and symmetry.