pure cubic field
A pure cubic field is an extension of ℚ of the form ℚ(3√n) for some n∈ℤ such that 3√n∉ℚ. If n<0, then 3√n=3√-|n|=-3√|n|, causing ℚ(3√n)=ℚ(3√|n|). 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 ℚ. For if n∈ℤ is cubefree with |n|≠1, then x3-n is its minimal polynomial
over ℚ. This polynomial
factors as (x-3√n)(x2+x3√n+3√n2) over K=ℚ(3√|n|). The discriminant
(http://planetmath.org/PolynomialDiscriminant) of x2+x3√n+3√n2 is (3√n)2-4(1)(3√n2)=3√n2-43√n2=-33√n2. Since the of x2+x3√n+3√n2 is negative, it does not factor in ℝ. Note that K⊆ℝ. 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 |