Rédei symbols and arithmetical mild pro-2-groups

Abstract

Abstract. Generalizing results of Morishita and Vogel, an explicit description of the triple Massey product for the Galois group GS(2) of the maximal 2-extension of Q unramified outside a finite set of prime numbers S containing 2 is given in terms of Rédei symbols. We show that mild pro-2-groups with Zassenhaus invariant 3 occur as Galois groups of the form GS(2). Furthermore, a non-analytic mild fab pro-2-group having only 3 generators is constructed .

Redei reciprocity, governing fields and negative Pell

“…Despite our additive definition, our proof of Rédei reciprocity in Section 8 views Rédei symbols in (46) as 'products' of local symbols [a, b, c] p , which are recognised in our key Lemma 8•1 as quadratic Hilbert symbols satisfying a well-known global product formula that we did not rename into a sum formula. On the other hand, our quadratic characters in (8) and biquadratic characters in (38) do take values in F 2 .…”

“…An immediate application of Rédei's reciprocity law in the form we have stated it is the existence of governing fields for the 8-rank of the narrow class group C(dp) of the quadratic field Q( √ dp), with d a fixed squarefree integer and p a variable prime. By this, we mean that there exists a normal number field 8,d with the property that for primes p, p d that are coprime to its discriminant and have the same Frobenius conjugacy class in Gal( 8,d /Q), the groups C(dp)/C(dp) 8 and C(dp )/C(dp ) 8 are isomorphic.…”

“…In the case of prime arguments a, b, c ≡ 1 mod 4, no dyadic ramification subtleties arise, and the symbol has been interpreted by Morishita [13, section 8•2] as an arithmetic Milnor invariant, leading to a description as a triple Massey product that is useful in the study of pro-2-extensions of Q with given ramification locus [8].…”

Higher Massey products in the cohomology of mild pro-p-groups

“…The question naturally arises whether mild pro-p-groups with Zassenhaus invariant ≥ 3 occur as arithmetical Galois groups of the form G S (p). For p = 2 a positive answer is given by the following two examples studied in [9], which actually satisfy the conditions given in 4.9:…”

Mild pro-$p$-groups and $p$-extensions of imaginary quadratic fields with non-trivial $p$-class group

“…Examples of groups G S which are mild are known for the field of rationals Q, and for imaginary quadratic fields not containing p-roots of unity with prime to p class number [20]. We can also mention the papers [8], [16], [1], [5] and [6]. Recently, in [13] Minač et al proved that if the maximal pro-p Galois group of a field is mild, then Positselski's and Weigel's Koszulity conjectures hold true for such a field.…”