Monogenic field

A monogenic field $𝐾$ is a number field possessing an Integral power basis, #m/def/num/alg i.e. the annoying index is unity.

Pure number field

Let $𝑝$ be a prime and $𝑎 \notin {0, 1, - 1}$ be a squarefree integer not divisible by $𝑝$ , and $𝜗 = 𝑝 \sqrt 𝑎$ . Then $𝐾 = O 𝐾$ is monogenic iff $𝑎 𝑝 - 1 ≢ 𝑝 2 1$ .¹

Proof

#missing/proof

Examples

Quadratic field, see Quadratic integers.
Cyclotomic field, see Cyclotomic integers.

#state/tidy | #lang/en | #SemBr

2022. Algebraic number theory course notes, §2.3.2, p. 45 ↩

Monogenic field

Pure number field

Examples

Footnotes