On the number of Pfister forms and the symbol length of fields with finite square class number

Let F be a field of characteristic not 2 with finitely many square classes.Using combinatorial arguments applied to objects related to vector spaces over finite fields, we deduce an upper bound for the number of Pfister forms over F .Moreover, we compute upper bounds for the n-symbol length of F (n N), i.e., the smallest integer sl n (F ) 0 such that to each quadratic form I n (F ) there exists some 0 k sl n (F ) and Pfister forms 1 , . . ., k such that 1 + + k mod I n+1 (F ).In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof.