Number field

Monogenic field

A monogenic field 𝐾 is a number field possessing an Integral power basis, alg i.e. the annoying index is unity.

Pure number field

Let 𝑝 be a prime and 𝑎 ∉{0,1, −1} be a squarefree integer not divisible by 𝑝, and 𝜗 =𝑝√𝑎. Then 𝐾 =O𝐾 is monogenic iff 𝑎𝑝−1 ≢𝑝21.¹

Proof

proof

Examples

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

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Footnotes

1. 2022. Algebraic number theory course notes, §2.3.2, p. 45 ↩