PlanetMath: pure cubic field

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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 , 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 .

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 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, has a root in but does not split completely in .

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.

"pure cubic field" is owned by Wkbj79.

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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 MSC:

11R16 (Number theory :: Algebraic number theory: global fields :: Cubic and quartic extensions)

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