Hilbert Symbol - Kaplansky Radical

Kaplansky Radical

The Hilbert symbol on a field F defines a map

where Br(F) is the Brauer group of F. The kernel of this mapping, the elements a such that (a,b)=1 for all b, is the Kaplansky radical of F.

The radical is a subgroup of F*/F*2, identified with a subgroup of F*. The radical contains is equal to F* if and only if F is not formally real and has u-invariant at most 2.