Algebraic number theory MOC

Cyclotomic field

A cyclotomic field is a number field obtained by adjoining a primitive th root of unity, alg i.e. , or equivalently, the splitting field of the separable polynomial

It follows that is a Finite Galois extension, with and degree given by the Euler totient function. The defining minimal polynomial of such a field is the so-called Cyclotomic polynomial.

• Its ring of integers are the Cyclotomic integers .
• See Group of roots of unity

This is especially well-behaved when is a prime power, see Prime power cyclotomic field.

Properties

1. The discriminant divides .¹

Proof of 1.

Since , it follows for some . Then

since . Taking the norm of both sides

where , proving ^P1.

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Footnotes

1. 2022. Algebraic number theory course notes, §2.4.1, p. 47 ↩