Witt Rings of Hasse Domains of Global Fields

We describe the structure of the Witt group W S(K) of any Hasse domain S(K) of a global field K by explicitly determining its cyclic direct summands and generators of each such summand. Using this description we find a sufficient condition for the existence of Witt ring isomorphism W S(K) W S′(L). We give also the necessary and sufficient condition for the existence of Witt ring isomorphism WK → WL of global fields K and L mapping the Witt ring W S(K) onto the Witt ring W S′(L).