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
continued fractions.) Find the continued fraction of
√about
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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