[[Number field]]
# Cubic field

A **cubic field** $K$ is a [[number field]] of degree 3, #m/def/num/alg 
i.e. $[K:\mathbb{Q}] = 3$,
where we call $K$ **pure** iff $K = \mathbb{Q}(d^{1/3})$ for some cubefree $d \in \mathbb{Z}$.

Contrary to the case for a [[quadratic field]], not every cubic field is pure.
Notably, a pure cubic field is necessarily complex,
so any real[^1] cubic field is impure.

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#state/develop | #lang/en | #SemBr

[^1]: In the sense of having only real embeddings.