Landau's theorem asserts that the asymptotic density of sums of two squares in the interval 1 <= n <= x is K/root logx where K is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals vertical bar n - x vertical bar <= x(is an element of) for a fixed is an element of and x -> infinity. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic f(0) is an element of F-q[T] of degree n and take is an element of with 1 > is an element of >= 2/n. Then the asymptotic density of polynomials f in the "interval" deg (f - f(0)) <= is an element of n that are of the form f = A(2) + TB2, A, B is an element of F-q [T] is 1/4(n) ([GRAPHICS]) as q -> infinity. This density agrees with the asymptotic density of such monic f's of degree n as q -> infinity, as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of f (-T-2), where f is a polynomial of degree n with a few variable coefficients: The Galois group is the hyperoctahedral group of order 2(n)n!.