From Common-Slot Chains to Heisenberg Central Products over Global Fields

Abstract

Let $F$ be a global field and let $p$ be an odd prime with $p\ne \mathrm{char}F$. Assuming first that ${\mu }_p\subset F$, we record in a uniform global-field form the length-four common-slot chain for equal degree-$p$ symbol classes and emphasize the signed normalization adapted to explicit norm constructions. Thus, from \[(a,b)_p=(c,d)_p\in\textrm{Br}(F)[p]\] one obtains $x,y\in F^{\times }$ such that \[(a,b)_p=(x^{-1},b)_p=(x,y)_p=(c^{-1},y)_p=(c,d)_p.\] For number fields, the underlying chain is the length-four chain lemma of Gille--Szamuely, based on Tate's simultaneous local--global theorem. The point developed here is that its signed form yields four compatible norm equations that can be used constructively. For the extraspecial central product $H_{p^3}\ast H_{p^3}$ over a $C_p^4$-Kummer extension, the central-embedding obstruction is $(a,b{)}_p-(c,d{)}_p$, while the four norm equations supplied by the chain assemble, under a natural independence hypothesis on the auxiliary Kummer classes, into an explicit factorized radical realization of the central product. Finally, when the ground field does not contain ${\mu }_p$, we show by a restriction--corestriction argument that the central-embedding obstruction is detected after passage to the cyclotomic extension $F({\mu }_p)$. Over that field the problem is Kummer, and the obstruction is again the difference of the two symbol classes.