Arithmetic of Quadratic Forms

has a solution in Fn. The representation problem of quadratic forms is to determine, in an effective manner, the set of elements of F that are represented by a particular quadratic form over F . We shall discuss the case when F is a field of arithmetic interest, for instance, the field of complex numbers C, the field of real numbers R, a finite field F, and the field of rational numbers Q. The representation problem for quadratic forms over any one of these fields has a very satisfactory solution. At the end, we shall discuss the solubility of equation (∗) over a subring R of F , that is, the problem of finding solution of (∗) in Rn. The most interesting and difficult case is when R is the ring of integers Z for which there is still a lot of questions left unanswered. Let f and g be two n-ary quadratic forms. We say that f and g are equivalent, written f ∼= g, if there exists an invertible matrix C ∈ GLn(F ) such that f(x) = g(Cx). This is the same as saying that there is an invertible homogeneous linear substitution of the variables x1, . . . , xn which takes the form g to the form f . Since