pure cubic field

A pure cubic field is an extension of of the form (n3) for some n such that n3. If n<0, then n3=-|n|3=-|n|3, causing (n3)=(|n|3). Thus, without loss of generality, it may be assumed that n>1.

Note that no pure cubic field is Galois (http://planetmath.org/GaloisExtension) over . For if n is cubefreeMathworldPlanetmath with |n|1, then x3-n is its minimal polynomialPlanetmathPlanetmath over . This polynomialPlanetmathPlanetmath factors as (x-n3)(x2+xn3+n23) over K=(|n|3). The discriminantMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/PolynomialDiscriminant) of x2+xn3+n23 is (n3)2-4(1)(n23)=n23-4n23=-3n23. Since the of x2+xn3+n23 is negative, it does not factor in . Note that K. Thus, x3-n has a root (http://planetmath.org/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.

Title pure cubic field
Canonical name PureCubicField
Date of creation 2013-03-22 16:02:19
Last modified on 2013-03-22 16:02:19
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 14
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11R16