Errata et Addenda to the First Printing of the Book Advanced Topics in Computational Number Theory by Henri Cohen (20000615 Version)

Graduate Texts in Mathematics 193, Springer-Verlag, 2000 First Printing 2000, XV + 578 pages. ISBN 0-387-98727-4 Springer-Verlag New York Berlin Heidelberg

p. 23 line 8, instead of “Corollary 1.2.11” read “Corollary 1.3.9” p. 31 line -11, instead of “𝔠_{j} = 𝔤_{n+1−j}𝔤_{n−j}^{−1}” read “𝔠_{j} = 𝔤_{j}𝔤_{j+1}^{−1}” p. 31 line -7, instead of “𝔤_{j}(A) = 𝔠_{n+1−j} ⋯ 𝔠_{n}” read “𝔤_{j}(A) = 𝔠_{j} ⋯ 𝔠_{n}” p. 44 line 3 of step 4 of Algorithm 1.7.3, exchange “a_{i,j}” with “a_{i,i}” (twice) p. 44 line 3 of step 7 of Algorithm 1.7.3, instead of “a_{j,i}A′_{j} − A′_{i}” read “A′_{j} −

a_{j,i}A′_{i}” and instead of “a_{j,i}V′_{j} − V′_{i}” read “V′_{j} − a_{j,i}V′_{i}”

p. 45 line 13, instead of “𝔠 = ∑_{i} 𝔞′_{i}𝔟′_{i}” read “𝔠 = ∑_{i} 𝔞′_{i}𝔟′_{i}^{−1}” p. 45 line 17, instead of “(n − i) × (n − i)” read “(n + 1 − i) × (n + 1 − i)” p. 45 line -7, instead of “step 5” read “step 2” p. 46 line 3 of Exercise 1, exchange “1/| N (x)|” and “| N (x)|” p. 60 line 10, instead of “θ_{1} = −R′(X)/R′_{Z}(X, k)” read “θ_{1} = −R′_{Z}(X, k)/R′(X)” p. 60 line 15, instead of “θ_{1}^{i}θ_{2}^{j}” read “θ_{1}^{i_{1}}θ_{2}^{i_{2}}” p. 62 line -5 and line -4, instead of “L_{1}(X)” read “L_{1}[X]” p. 65 line -8, instead of “∑_{0≤j<n_{1}} c_{i,j}a_{j} = b_{i}” read “∑_{0≤j<n_{1}} c_{i,j}b_{j} = a_{i}” p. 65 line -3 (twice) and line -1 (once), instead of “α” read “θ_{2}” p. 66 line 3 of step 2 of Algorithm 2.1.12, instead of “(1, 0, …, 0)^{t}” read

“(0, 1, 0, …, 0)^{t}”

p. 85 line 8, instead of “α_{j}” read “γ_{j}” p. 88 line 1 of Lemma 2.3.7, instead of “α ∈ K” read “α ∈ L” p. 88 line -12, instead of “N_{L/K}(I) ∈ I” read “N_{L/K}(I) ⊂ I” p. 92 lines 8 and 9, exchange “𝔓 is unramified” with “𝔓 is ramified” p. 99 Replace steps 3 and 4 of Algorithm 2.3.21 by the following:

3. [Terminate] Using Algorithm 1.5.1, compute an HNF pseudo-basis (γ_{i}, 𝔠_{i}) for the intersection I^{−1} ← I_{1} ∩ I_{2}.

p. 106 line -1, instead of “ℤ_{K} ∖ 𝔭^{−1}” read “𝔭^{−1} ∖ ℤ_{K}” p. 110 lines 5 and 6 of Algorithm 2.4.12, replace twice “∑_{1≤k≤n}” by “∑_{0≤k≤n}” p. 114 line 19, instead of “θ ∈ K” read “θ_{1} ∈ K” p. 131 Exercise 33, instead of “proof given in” read “proof of” p. 131 Exercise 34, replace twice “t_{i}” by “T_{i}”, and replace four times “r_{i}” by

“R_{i}”

p. 131 Exercise 36, instead of “ω^{2}(ω + 3)” read “ω^{2}(−ω − 3)” p. 135 line -5, instead of “ρ(α) = (\overline{α}, (sign(σ_{i}(α))_{σ_{i}∈𝔪_{∞}}))” read “ρ(α) =

(\overline{α}, (sign(σ_{i}(α)))_{σ_{i}∈𝔪_{∞}})”

p. 161 Exercise 7 d), instead of “necessary” read “necessarily” p. 161 Exercise 15 a), add the property “𝔡(L/K) | 𝔪_{0}^{ℓ^{r}−1}” p. 187 line 1, instead of “ϕ(\overline{β}) = \overline{β} , and” read “ϕ(\overline{β}) = \overline{β}, and”

p. 201 line 3 of step 2 of Algorithm 4.2.16, instead of “β ← β ∏_{i}(1 + γ_{a,i}^{−y_{a,i}})” read β ← β ∏_{i}(1 + γ_{a,i})^{−y_{a,i}}”

p. 212 line -14, instead of “(ℤ/𝔪)^{∗}” read “(ℤ_{K}/𝔪)^{∗}” p. 221 Exercise 18 c), instead of “prime ideal below” read “prime number

below”

p. 226 line 9, instead of “field K, a congruence” read “field K and a congruence”

p. 230 middle, instead of “the next section” read “Section 5.2.4” p. 255 line -3, instead of “γ ∈ K^{∗}^{ℓ}” read “γ ∈ K^{∗}” p. 260 line -10, instead of “Thus,” read “Hence,” p. 293 Exercise 4 a), instead of “exists” read “exist” (twice) p. 308 line 3, instead of “(−i)^{f_{∞}}” read “(−i)^{|𝔣_{∞}|}” p. 308 step 5 of Algorithm 6.2.4, instead of “(−i)^{|f_{∞}|}” read “(−i)^{|𝔣_{∞}|}” p. 315 middle, instead of “elliptic function j(τ)” read “modular function j(τ)”

p. 318 middle, instead of “(\frac{2iπ}{ω_{2}} )^{24}” read “(\frac{2iπ}{ω_{2}} )^{12}”

p. 345 Remove Exercise 8 (it is Exercise 15 d) p. 346 Exercise 25, put the parenthetical statement “(where as usual we set

𝔣 ∩ ℤ = fℤ)” at the end of the sentence, without parentheses

p. 425 line 2 of step 9, instead of “d_{3} ← ⌈(b − s)/2⌉ − 1, d_{4} ← ⌊(b + s)/2⌋ + 1” read “d_{3} ← ⌈(b − s)/2⌉, d_{4} ← ⌊(b + s)/2⌋”

p. 434 line 11, instead of “α/α′” read “(α/α′)ℤ_{K}” p. 434 line -13, instead of “whose prime factors are only prime ideals above 2”

read “dividing 2”

p. 439 line 2 of (4), instead of “they will” read “it will” p. 439 line 5 of (4), instead of “repeat” read “recompute” p. 439 middle, instead of “the case of prime degree” read “the prime degree

case”

p. 444 line -8, instead of “G_{18}^{+}, G_{18}^{−}, G_{36}, and G_{72}” read “G_{36}^{+}, G_{36}^{−}, and G_{72}” p. 451 line -3, instead of “Schwartz’s” read “Cauchy–Schwarz’s” p. 455 line 1, instead of “Schwartz’s” read “Cauchy–Schwarz’s” p. 469 replace Exercise 6 b) by the following: “Let R_{1} denote an even integer

such that 0 ≤ R_{1} ≤ 2r_{1}. Show that the number of K-isomorphism classes of quadratic extensions L/K with N_{K/ℚ}(𝔡(L/K)) ≤ x and R_{1} real embeddings is asymptotic to (\binom{r_{1}}{R_{1}/2}/2^{r_{1}})Q_{K} · x, where Q_{K} is as above.”

p. 470 line 1 of Exercise 14, replace “ and” by “, ” p. 471 Exercise 16 a), instead of “Schwartz’s” read “Cauchy–Schwarz’s” p. 471 Exercise 16 a), instead of “show that” read “prove Lagrange’s identity” p. 483 line -6, instead of “θ_{k}(τ)(θ_{0}(σ)^{k} − 1) = 0” read “(θ_{0}(σ)^{k} − 1)θ_{k}(τ) = 0” p. 484 line -12, instead of “𝔇(L/K) = ℤ_{L}^{∗}” read “ℤ_{L}^{∗} = 𝔇(L/K)^{−1}” p. 488 middle, instead of “v_{𝔓}(𝔣_{2}) is even” read “v_{𝔓}(𝔣_{2}) is even when 𝔭 is

ramified in K_{2}/K”

p. 520 rewrite Exercise 12 as follows: “Let L/K be an extension of number fields, let P be a monic polynomial with coefficients in K, let 𝔭 be a prime ideal of K, and let α ∈ L be a root of P. Assume that the 𝔭-adic valuations of all the coefficients of P are nonnegative. Show that v_{𝔓}(α) ≥ 0 for any prime ideal 𝔓 of L above 𝔭.”

p. 520 line 1 of Exercise 15, instead of “let ℓ is a prime number” read “ℓ a prime number”

p. 558 add at the appropriate place “Γ_{∞}: group of integer translations, 122” p. 573 add at the appropriate place “Lagrange’s identity, 471”