2012/04/09 by Ming-chang Kang, Kang, Ming-chang
Computer Science · Mathematics · #12F12 #13A50 #14E08 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1204.1796
openalex publication_date 2012/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be any field, G be a finite group acting on the rational function field k(xg:g∈ G) by h⋅ xg=xhg for any h,g∈ G. Define k(G)=k(xg:g∈ G)G. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether's problem. We prove that, if G is a Frobenius group with abelian Frobenius kernel, then k(G) is retract k-rational for any field k satisfying some mild conditions. As an application, we show that, for any algebraic number field k, for any Frobenius group G with Frobenius complement isomorphic to SL2(\bmF5), there is a Galois extension field K over k whose Galois group is isomorphic to G, i.e. the inverse Galois problem is valid for the pair (G,k). The same result is true for any non-solvable Frobenius group if k(ζ8) is a cyclic extension of k.