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单词 PowerBasisOvermathbbZ
释义

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.)

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更新时间:2025/5/4 12:42:55