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

+