Math 346/646 - Homework 8 hints
1. This should work very similarly to the proof of Lemma 2 (and in particular, in the use of
the triangle inequality.)
2. This requires working explicitly through the proof of Lemma 3 in some detail in the case
that φ(x) = x4 + 4x2 + 4 and ψ(x) = 4x3 − 8x. Find polynomials F (x) and G(x) so that
128F (x)φ(x) + 128G(x)ψ(x) = 128
and 128F (x) and 128G(x) are polynomials with integer coefficients. Conclude that if gcd(m, n) =
1, then gcd(Φ(m, n), Ψ(m, n)) divides 128.
Show that
|φ(x)|+|ψ(x)|
max{|x|4 ,1}
≥ 1.
3. (a) The identity is (1, 0). The inverse of (x, y) is (x, −y). The associative law probably
requires some careful algebra.
(b) Let S be a finite set of prime numbers, and let HS (Q) denote the set of points in H(Q)
where the denominators of the x and y coordinates only involve primes in S. (For example,
(3/5, 4/5) ∈ H{5} (Q).)
Show that HS (Q) is a subgroup of H(Q).
Assume that P1 , . . ., Pr generate H(Q). Show that there is a set S so that Pi ∈ HS (Q) for all
S. Conclude that HS (Q) = H(Q).
Given any finite set S of primes, construct an point P ∈ H(Q) (using the parametrization of
x2 + y 2 = 1 on page 11 of the text) for which P 6∈ HS (Q) and hence HS (Q) 6= H(Q). This
contradiction shows that H(Q) is not finitely generated.
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