Solutions to Problems 3.5, 3.9.5, 3.11, 3.13.5

Next, we claim that x ∈ Qp is a square if and only if x = py for n ∈ Z, and y ∈ Zp a padic unit. One direction is obvious. The second direction is also fairly obvious–just refer to the previous case, and also note that p is itself NOT a square in Qp (since if it had a square root, then that square root would have valuation 12 ). We can conclude then that [ Qp : Q×2 p ] = 4, and is given by {1, p, u, pu} for u a quadratic non-residue in Zp . Referring to the text below this Exercise in the course notes, we can moreover conclude that Qp /(Q×2 p ) ∼= Z/2Z× Z/2Z 3.9.5 A very special and important quadratic form is qH(x1, x2) = x1x2, the so-called hyperbolic plane. a) Let K be any field of characteristic different from 2. Give an explicit change of variables that diagonalizes qH. Solutions: Short answer: let