ABELIAN-BY-CENTRAL GALOIS GROUPS OF FIELDS I: A FORMAL DESCRIPTION

作者：Topaz Adam^*

来源：Transactions of the American Mathematical Society, 2017, 369(4): 2721-2745.

DOI：10.1090/tran/6740

• Abelian-by-central
• Galois groups
• group cohomology
• profinite groups

Let K be a field whose characteristic is prime to a fixed positive integer n such that mu(n). K, and choose omega mu(n) as a primitive n-th root of unity. Denote the absolute Galois group of K by Gal(K), and the mod-n central-descending series of Gal(K) by Gal(K)((i)). Recall that Kummer theory, together with our choice of omega, provides a functorial isomorphism between Gal(K)/Gal(K)(2) and Hom(Kx, Z/n). Analogously to Kummer theory, in this note we use the Merkurjev-Suslin theorem to construct a continuous, functorial and explicit embedding Gal(K)(2)/Gal(K)(3) . Fun(K {0, 1}, (Z/n) 2), where Fun(K {0, 1}, (Z/n) 2) denotes the group of (Z/n) 2-valued functions on K {0, 1}. We explicitly determine the functions associated to the image of commutators and n-th powers of elements of Gal(K) under this embedding. We then apply this theory to prove some new results concerning relations between elements in abelian-by-central Galois groups.

• 出版日期2017-4