pure cubic field
Note that no pure cubic field is Galois (http://planetmath.org/GaloisExtension) over . For if is cubefree with , then is its minimal polynomial over . This polynomial factors as over . The discriminant (http://planetmath.org/PolynomialDiscriminant) of is . Since the of is negative, it does not factor in . Note that . Thus, has a root (http://planetmath.org/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.
|Title||pure cubic field|
|Date of creation||2013-03-22 16:02:19|
|Last modified on||2013-03-22 16:02:19|
|Last modified by||Wkbj79 (1863)|