Errata, Changes, and Addenda for the Books “Number Theory”
Volumes I and II
by Henri Cohen
Last modified: November 30, 2008
Errata: errors, mathematical or otherwise, which must be corrected.
Changes: modifications which improve the presentation.
Addenda: additions of relevant, useful, and/or interesting material.
Errata:
In Volume I:
p. 4 line 9 and 19, replace “finite (this can...exercise).” by “finite.”
p. 13 line -11, replace “V ∈ dkA1” by “V = dkA1”
p. 37 middle, add an “=” sign between
(
a
|m|
)
and sign(a+ km)
p. 44 line 2, replace “n = 1 + |D|/2” by “n = 1 +D/2”
p. 49 lines -3 to -1, replace “, and one can... Exercise 17).” by “; see also
Exercise 17.”
p. 55 line -11, replace “Elkies et al.” by “Cohn et al.”
p. 80 line 7, add a white square at the end of the proof of Lemma 2.5.13
p. 85 line -16, replace “a+ b ≡ p− 2 (mod 9)” by “a+ b ≡ q− 2 (mod 9)”
p. 86 line 5, replace “a2 − ab+ b2 = p” by “a2 − ab+ b2 = q”
p. 95 replace Exercise 17 (which is false) by the following (hopefully correct)
exercise:
“17.
(a) Show that for any one of the four primitive characters χ modulo 16, the
root number W (χ) is a 16th root of unity.
(b) Assume that m > 3 and m
6≡ 2 (mod 4). Show that if W (χ) is a root of
unity for all primitive characters χ modulo m, then in fact W (χ)m = 1,
and furthermore that λ(m) | m, where λ(m) is Carmichael’s function as
defined in Exercise 6 (I do not know how to prove this, but it must be
true and not too difficult).”
p. 179 in Exercise 4, replace “f(x+ a2) = f(x) + f(a)2” by
“σ(x+ a2) = σ(x) + σ(a)2”
p. 210 lines 3 to 13, replace “vp(... as claimed” by the following:
1
“
vp
(
n!
(
a
n
))
≥
∑
0≤k<q
k
∑
0≤i<n
vp(i)=k
1 + vp(a)
∑
0≤i<n
vp(i)≥q
1
=
∑
0≤k<q
k(dn/pke − dn/pk+1e) + vp(a)dn/pqe
=
∑
1≤k≤q
dn/pke − θdn/pqe
again by Abel summation. Transforming the ceiling into the floor function it is
clear that this gives
vp
(
n!
(
a
n
))
≥ max(q − θ − vp(n), 0)−
∑
1≤k≤q
bn/pkc − θbn/pqc ,
and since by (1) we have vp(n!) =
∑
k≥1bn/pkc, it follows that
vp
((
a
n
))
≥ max(vp(a)− vp(n), 0)− θbn/pqc −
∑
k