[[Number field]]
# Monogenic field

A **monogenic field** $K$ is a [[number field]] possessing an [[Bases for a number field#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 $\vartheta = \sqrt[p]{ a }$.
Then $K = \mathcal{O}_{K}$ is monogenic iff $a^{p-1} \not\equiv_{p^2} 1$.[^2022]

> [!missing]- Proof
> #missing/proof


  [^2022]: 2022\. [[Sources/@bakerAlgebraicNumberTheory2022|Algebraic number theory course notes]], §2.3.2, p. 45

## Examples

- [[Quadratic field]], see [[Quadratic integers]].
- [[Cyclotomic field]], see [[Cyclotomic integers]].

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