Galois group of a biquadratic extension

This article proves that biquadratic extensions correspond precisely to Galois extensionsMathworldPlanetmath with Galois groupMathworldPlanetmath isomorphicPlanetmathPlanetmathPlanetmath to the Klein 4-group V4 (at least if the characteristicPlanetmathPlanetmath of the base fieldMathworldPlanetmathPlanetmath is not 2). More precisely,

Theorem 1.

Let F be a field of characteristic 2 and K a finite extensionMathworldPlanetmath of F. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    K=F(D1,D2) for some D1,D2F such that none of D1,D2, or D1D2 is a square in F.

  2. 2.

    K is a Galois extension of F with Gal(K/F)V4;

Proof. Suppose first that condition (1) holds. Then [F(D1):F]=[F(D2):F]=2 since neither D1 nor D2 is a square in F. Now obviously


and so [K:F(D1)]2. If K=F(D1), then D2F(D1), so D2=a+bD1 and D2=a2+b2D1+2abD1. Thus a=0 or b=0. If b=0, then D2 is a square. If a=0, then D1D2=b2D12 is a square. In any case, this is a contradictionMathworldPlanetmathPlanetmath. Thus K is a quadratic extension of F(D1). So [K:F]=4. But K is the splitting fieldMathworldPlanetmath for (x2-D1)(x2-D2), since the splitting field must contain both square roots, and the polynomialMathworldPlanetmathPlanetmathPlanetmath obviously splits in K, so G=Gal(K/F) has four elements

id σ:{D1-D1D2D2 τ:{D1D1D2-D2 στ:{D1-D1D2-D2

and is thus isomorphic to V4.

Now assume that condition (2) holds. Since Gal(K/F)V4, there must be three intermediate subfieldsMathworldPlanetmath E1,E2,E3 between F and K of degree 2 over F corresponding to the three subgroupsMathworldPlanetmathPlanetmath of V4 of order 2. Thus each of these is a quadratic extension. Suppose E1=F(D1),E2=F(D2) where neither D1 nor D2 is a square in F. The fact that E1E2 implies as above that D1D2 is also not a square in F (in fact E3=F(D1D2). Thus E1E2E1,E2, and is of degree 4 over F, so K=E1E2=F(D1,D2).

Title Galois group of a biquadratic extension
Canonical name GaloisGroupOfABiquadraticExtension
Date of creation 2013-03-22 17:44:06
Last modified on 2013-03-22 17:44:06
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Theorem
Classification msc 11R16