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Math 346/646 - Homework 3 Assigned: 9/3. Due: 9/10 at the start of class. Notation: Exercise a.b stands for exercise b in Chapter a of Silverman and Tate. Note that the exercises are all together at the end of the chapter. A problem marked with a ∗ is a challenge problem. Problems: 1. Let G = {r ∈ R : r 6= −1}. Define a new binary operation on G by x ∗ y = x + y + xy. (For example, 3 ∗ 4 = 19.) (a) Prove that ∗ makes G into a group. (b) Let R× be the group of nonzero real numbers under multiplication. Show that the function φ : G → R× defined by φ(r) = r + 1 is an isomorphism. 2. Exercise A.1 from the textbook. 3. ∗ Let y 2 = x3 + Ax + B be an elliptic and assume that the cubic x3 + Ax + B has three distinct real roots: r1 < r2 < r3 . Let G = E(R) and let H = {(x, y) ∈ E(R) : x ≥ r3 } together with the point at infinity. Prove that H is a subgroup of G. (You may assume that G is a group. Hint: The other part of E(R) is K = {(x, y) ∈ E(R) : r1 ≤ x ≤ r2 }. Could a line intersect K in exactly one point, if it wasn’t tangent at that point?) 1