2.12. Exercises 69 2.12. Exercises Exercise 2.12.1. Let f(x) = a0xn + a1xn−1 + . . . + an, with ai ∈ Z. Prove that if x = p q ∈ Q, with gcd(p, q) = 1, is a solution of f(x) = 0, then an is divisible by p and a0 is divisible by q. Exercise 2.12.2. Let C be the conic defined by x2 − 2y2 = 1. (1) Find all the rational points on C. (Hint: the point O = (1,0) belongs to C. Let L(t) be the line that goes through O and has slope t. Since C is a quadratic and L(t) ∩ C contains at least one rational point, there must be a second point of intersection Q. Find the coordinates of Q in terms of t.) (2) Let α = 1+ √ 2. Calculate α2 = a + b √ 2 and α4 = c + d √ 2 and verify that (a, b) and (c, d) are integral points on C : x2 − 2y2 = 1. (Note: in fact, if α2n = e + f √ 2, then (e, f) ∈ C and the coeﬃcients of α2n+1 are a solution of x2 − 2y2 = −1.) (3) (This problem is only for those who already know about continued fractions.) Find the continued fraction of √ 2 and find the first 6 convergents. Use the convergents to find three distinct (positive) integral solutions of x2 − 2y2 = 1, other than (1,0). (Note: the reader should remind herself or himself how to find the continued fraction and conver- gents by hand, then check his or her answer using Sage see Appendix A.4.) Exercise 2.12.3. Let C/Q be an aﬃne curve. (1) Suppose that C/Q is given by an equation of the form C : xy2 + ax2 + bxy + cy2 + dx + ey + f = 0. (2.13) Find an invertible change of variables that takes the equa- tion of C onto one of the form xy2+gx2+hxy+jx+ky+l = 0. (Hint: consider a change of variables X = x + λ, Y = y). (2) Suppose that C /Q is given by an equation of the form C : xy2 + ax2 + bxy + cx + dy + e = 0. (2.14) Find an invertible change of variables that takes the equa- tion of C onto one of the form y2 + αxy + βy = x3 + γx2 +

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