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单词 BiquadraticExtension
释义

biquadratic extension


A biquadratic extension of a field F is a Galois extensionMathworldPlanetmath K of F such that Gal(K/F) is isomorphicPlanetmathPlanetmathPlanetmath to the Klein 4-group. It receives its name from the fact that any such K is the compositum of two distinct quadratic extensions of F. The name can be somewhat misleading, however, since biquadratic extensions of F have exactly three distinct subfieldsMathworldPlanetmath that are quadratic extensions of F. This is easily seen to be true by the fact that the Klein 4-group has exactly three distinct subgroupsMathworldPlanetmathPlanetmath of order (http://planetmath.org/OrderGroup) 2.

Note that, if α,βF, then F(α,β) is a biquadratic extension of F if and only if none of α, β, and αβ are squares in F.

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更新时间:2025/5/5 0:02:45