Galois Theory and Noether's Problem Meredith Blue ... - MAA Sections

andσ(x 0 x 1 ) = σ(x 0 )σ(x 1 ) = x 1 x 0 = x 0 x 1It is easy to see that x 0 + x 1 and x 0 x 1 are transcendental over Q so thereexists a Galois extension with group G over Q. Of course, we saw a concreteexample earlier: Q( √ 2)/Q.Of course, Q( √ 3)/Q is also a Galois extension over Q with group G. BothQ( √ 2)/Q and Q( √ 3)/Q can be realized as “specializations” of Q(x 0 , x 1 )/Q(x 0 +x 1 , x 0 x 1 ) by using the following map.φ : Q(x 0 , x 1 ) → Q( √ a)is a field isomorphism (i.e. φ(xy) = φ(x)φ(y) and φ(x + y) = φ(x) + φ(y))satisfying:φ(q) = q ∀q ∈ Q and φ(x 0 ) = √ a and φ(x 1 ) = − √ a.In particular φ(x 0 x 1 ) = φ(x 0 )φ(x 1 ) = −a ∈ Q, and φ(x 0 + x 1 ) = φ(x 0 ) +φ(x 1 ) = 0 ∈ Q, so Q(x 0 + x 1 , x 0 x 1 ) ∼ = Q. In this specialization a can beany rational number which is not a square. The parameterization can beexpressed by the following sentence: “Any extension of Q with a 2-elementGalois group is obtained by adding a root of the polynomial x 2 − a where ais any rational number such that √ a is not rational.”References[H] Herstein, I. N. Topics in Algebra. Xerox Corporation. 1975[I][N][S6][Se]Issacs, I. Martin. Abstract Algebra. Brooks/Cole Publishing Company.1994.Noether, E. “Gleichungen mit Vorgeschribener Gruppe.” Math Ann.,78 221-229. (1918)Saltman, David J. “Groups acting on fields, Noether’s Problem.”Contemporary Mathematics. Vol. 43 267-277. (1985)Serre, Jean-Peirre. Topics in Galois Theory. Jones and Bartlett Publishers,Inc. 1992.8