power basis over

Let K be a number field with [K:]=n and 𝒪K denote the ring of integersMathworldPlanetmath of K. Then 𝒪K has a power basis over (sometimes shortened simply to power basis) if there exists αK such that the set {1,α,,αn-1} is an integral basis for 𝒪K. An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) condition is that 𝒪K=[α]. Note that if such an α exists, then α𝒪K and K=(α).

Not all rings of integers have power bases. (See the entry biquadratic field for more details.) On the other hand, any ring of integers of a quadratic field has a power basis over , as does any ring of integers of a cyclotomic fieldMathworldPlanetmath. (See the entry examples of ring of integers of a number field for more details.)

Title power basis over
Canonical name PowerBasisOvermathbbZ
Date of creation 2013-03-22 15:56:55
Last modified on 2013-03-22 15:56:55
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 17
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11R04
Synonym power basis
Synonym power bases
Related topic ConditionForPowerBasis