Cyclotomic extension

Builds on Field extension Primitive root of unity Root of unity Splitting field

Definition

Let $K$ be a field and let $n \ge 1$ be an integer that is not divisible by the characteristic of $K$ . A field extension $L/K$ is an $n$ -th cyclotomic extension if both conditions hold:

$L$ contains a primitive $n$ -th root of unity $\zeta_n$ ;
$L = K(\zeta_n)$ .

Equivalently, $L$ is a splitting field of $x^n - 1$ over $K$ . More generally, let $S$ be a set of positive integers. Then $L/K$ is an $S$ -cyclotomic extension if $L$ contains a primitive $n$ -th root of unity for each $n \in S$ . Moreover $L$ must be generated over $K$ by the roots of unity of these orders.

Remarks

Example: $\Q(i)/\Q$ is a $4$ -th cyclotomic extension, since $i$ has order $4$ in $\C^\times$ and $\Q(i)$ is generated by $i$ . Non-example: $\Q(\sqrt[3]{2})/\Q$ is not a $3$ -rd cyclotomic extension, since $\Q(\sqrt[3]{2})$ is a subfield of $\R$ and contains no primitive cube root of unity. The two forms agree because the roots of $x^n - 1$ are the powers of $\zeta_n$ . An $n$ -th cyclotomic extension is Galois, and $\sigma \mapsto k$ with $\sigma(\zeta_n) = \zeta_n^k$ is an injective homomorphism $\Gal(L/K) \to (\Z/n\Z)^\times$ . For $K = \Q$ this map is an isomorphism, and $L$ is the cyclotomic field $\Q(\zeta_n)$ .

Reading path

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Complex numbers object
Group concept
Root of unity definition
Ring definition
Primitive root of unity definition
Commutative ring definition
Polynomial ring definition
Subring definition
Unit of a ring definition
Polynomial evaluation definition
Field definition
Root of a polynomial definition
Subfield definition
Field extension definition
Split polynomial definition
Intermediate field definition
Field generated by adjoining elements definition
Splitting field definition
Cyclotomic extension definition

Tag	`TWG8QKY8TA`
URI	`m2://TWG8QKY8TA`
Kind	definition
Writing	W0 stub
MSC	12F10, 11R18
Defines	Cyclotomic extension
Uses	Field extension, Primitive root of unity, Root of unity, Splitting field
Changed	2 November 2026

Cyclotomic extension

Definition

Remarks

Reading path

Further reading

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