Landau's theorem asserts that the asymptotic density of sums of two squares in the interval is , where is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals for a fixed and . This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic of degree and take with . Then the asymptotic density of polynomials in the" interval" that are of the form , is ${1\over 4^ n}\big (\matrix {2n\atop n}\big) $ as . This density agrees with the asymptotic density of such monic 's of degree as , as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of , where is a polynomial of degree with a few variable coefficients: The Galois group is the hyperoctahedral group of order .