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pure cubic field (Definition)

A pure cubic field is an extension of $ \mathbb{Q}$ of the form $ \mathbb{Q}(\sqrt[3]{n})$ for some $ n \in \mathbb{Z}$ such that $ \sqrt[3]{n} \notin \mathbb{Q}$. If $ n<0$, then $ \sqrt[3]{n}=\sqrt[3]{-\vert n\vert}=-\sqrt[3]{\vert n\vert}$, causing $ \mathbb{Q}(\sqrt[3]{n})=\mathbb{Q}(\sqrt[3]{\vert n\vert})$. Thus, without loss of generality, it may be assumed that $ n>1$.

Note that no pure cubic field is Galois over $ \mathbb{Q}$. For if $ n \in \mathbb{Z}$ is cubefree with $ \vert n\vert \neq 1$, then $ x^3-n$ is its minimal polynomial over $ \mathbb{Q}$. This polynomial factors as $ (x-\sqrt[3]{n})(x^2+x\sqrt[3]{n}+\sqrt[3]{n^2})$ over $ K=\mathbb{Q}(\sqrt[3]{\vert n\vert})$. The discriminant of $ x^2+x\sqrt[3]{n}+\sqrt[3]{n^2}$ is $ \left( \sqrt[3]{n} \right)^2-4(1)\left( \sqrt[3]{n^2} \right)=\sqrt[3]{n^2}-4\sqrt[3]{n^2}=-3\sqrt[3]{n^2}$. Since the discriminant of $ x^2+x\sqrt[3]{n}+\sqrt[3]{n^2}$ is negative, it does not factor in $ \mathbb{R}$. Note that $ K \subseteq \mathbb{R}$. Thus, $ x^3-n$ has a root in $ K$ but does not split completely in $ K$.

Note also that pure cubic fields are real cubic fields with exactly one real embedding. Thus, a possible method of determining all of the units of pure cubic fields is outlined in the entry regarding units of real cubic fields with exactly one real embedding.



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Cross-references: units of real cubic fields with exactly one real embedding, units, real embedding, fields, real, negative, factors, polynomial, minimal polynomial, cubefree, without loss of generality, extension
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This is version 11 of pure cubic field, born on 2006-06-26, modified 2007-06-26.
Object id is 8088, canonical name is PureCubicField.
Accessed 2066 times total.

Classification:
AMS MSC11R16 (Number theory :: Algebraic number theory: global fields :: Cubic and quartic extensions)

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