Number field

Cubic field

A cubic field 𝐾 is a number field of degree 3, alg i.e. [𝐾 :ℚ] =3, where we call 𝐾 pure iff 𝐾 =ℚ(𝑑1/3) for some cubefree 𝑑 ∈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 real¹ cubic field is impure.

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Footnotes

1. In the sense of having only real embeddings. ↩