Errata et Addenda to the First Printing of the Book Advanced Topics in Computational Number Theory by Henri Cohen (20000211 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. 131 Exercise 33, instead of “proof given in” read “proof of” 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. 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”