Let p be a fixed odd prime and k an algebraic extension of Z pZ such that k ∗ the multiplicative group of k, is 2-divisible. Let x be a fixed element, transcendental over k. Let π be either 1 x or a prime in k[ x], and F π the completion of k( x) with respect to the discrete rank 1 valuation that π gives rise to. Let O π be the ring of integers of F π . Consider the integral quadratic form Q( X, Y) = aX 2 + bXY + cY 2, a, b, c ∈ k( x), and X and Y taking values in k[ x]. Two such forms are said to be in the same genus if they are equivalent over O π for all π. They are in the same class if they are equivalent over k[ x]. The class number h Q of such a form is the number of classes in a genus. We prove the following: All forms of the above type with a, b, c ∈ k[ x], ( a, b, c) = 1, b 2 − 4 ac = Δ with fixed Δ (up to multiplication by a constant) are in the same genus. If deg Δ ≤ 2, h Q = 1. If deg Δ ≥ 3, h Q = ∞.