Quadratic Reciprocity in Characteristic 2

Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] lets us decide whether or not a quadratic congruence f ≡ x2 mod π is solvable, where the modulus π is irreducible in F[T ] and f 6≡ 0 mod π. This is similar to the quadratic reciprocity law in Z. We want to develop an analogous reciprocity law when F has characteristic 2. At first it does not seem that there is an analogue: when F has characteristic 2, every element of the finite field F[T ]/π is a square, so the congruence f ≡ x2 mod π is always solvable (and uniquely, at that). This is uninteresting. The correct quadratic congruence to try to solve in characteristic 2 is