This paper is concerned with the connections between the Witt ring of a field and the structure of certain Galois extensions of that field. In particular, it is shown that the Witt ring determines, and is determined by, the Galois group of a certain 2-extension of the field (with an unavoidable uncertainty over the characteristic of the Witt ring in the case when the form \(\langle 1,1 \rangle\) is universal). This group is called the \(W\)-group of the field \(F\), denoted \({\mathcal G}_F\). It can be viewed as the Galois group over \(F\) of the compositum over \(F\) of all quadratic, cyclic of order 4, and dihedral of order 8 extensions. \(W\)-groups are calculated for several specific fields and classes of fields, and all possible abelian \(W\)-groups are determined. An alternative way of computing \(W\)-groups, using cohomological methods, is given. Finally, the work on \(W\)-groups is put in the context of Merkurjev's result on the isomorphism from \(k_2F\) to \(Br_2F\).