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Let k be a field and V be an algebraic fc-scheme of dimension w—1. V is called a Severi-Brauer /;-scheme if there exists a separable algebraic extension L/k such that V x kL and Pw_1(L)=Proj L(XX, • • • , Xn) are isomorphic as /.-schemes (7). V is said to be split by L/k. V is called a trivial Severi-Brauer ^-scheme if V and -Pw_i(fc) are isomorphic as fc- schemes. Let K/k be a finite Galois extension and let G=Gal(X/A:). The iso­ morphism classes (as /^-schemes) of Severi-Brauer /c-schemes of dimension n—\ which are split by K/k are in canonical one-one correspondence with the elements of the cohomology set H^G.PGLin.K)) (7). The exact sequence l->K*-*GL(n, K)-+PGL(n, K)-+\ induces a map

The zeta function of a class of rational varieties