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\centerline{\bf Errata et Addenda to the First Printing of the Book}
\smallskip
\centerline{\bf Advanced Topics in Computational Number Theory}
\smallskip
\centerline{by \bf Henri Cohen}
\smallskip
\centerline{(20000211 Version)}
\bigskip

\medskip
{\obeylines
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
\bigskip
p. 131\quad Exercise 33, instead of ``proof given in'' read ``proof of''
p. 161\quad Exercise 7 d), instead of ``necessary'' read ``necessarily''
p. 161\quad Exercise 15 a), add the property ``$\gd(L/K)\mid\m_0^{\ell^r-1}$''
p. 187\quad line 1, instead of ``$\phi\bigl(\ov{\be}\bigr)=\ov{\be}$ , and'' read ``$\phi\bigl(\ov{\be}\bigr)=\ov{\be}$, and''
p. 212\quad line -14, instead of ``$(\Z/\m)^*$'' read ``$(\Z_K/\m)^*$''
p. 221\quad Exercise 18 c), instead of ``prime ideal below'' read ``prime number below''
p. 226\quad line 9, instead of ``field $K$, a congruence'' read ``field $K$ and a congruence''
p. 230\quad middle, instead of ``the next section'' read ``Section 5.2.4''
p. 255\quad line -3, instead of ``$\gamma\in {K^*}^\ell$'' read ``$\gamma\in K^*$''
p. 260\quad line -10, instead of ``Thus,'' read ``Hence,''
p. 293\quad Exercise 4 a), instead of ``exists'' read ``exist'' (twice)
p. 308\quad line 3, instead of ``$(-i)^{f_\infty}$'' read ``$(-i)^{\abs{\f_\infty}}$''
p. 308\quad step 5 of Algorithm 6.2.4, instead of ``$(-i)^{\abs{f_\infty}}$'' read ``$(-i)^{\abs{\f_\infty}}$''
p. 315\quad middle, instead of ``elliptic function $j(\tau)$'' read ``modular function $j(\tau)$''
p. 318\quad middle, instead of ``$\left(\dfrac{2i\pi}{\omega_2}\right)^{24}$'' read ``$\left(\dfrac{2i\pi}{\omega_2}\right)^{12}$''
p. 345\quad Remove Exercise 8 (it is Exercise 15 d)
p. 346\quad Exercise 25, put the parenthetical statement ``(where as usual we set $\f\cap\Z=f\Z$)'' at the end of the sentence, without parentheses
p. 425\quad line 2 of step 9, instead of ``$d_3\gets\lceil(b-s)/2\rceil-1$, $d_4\gets\lfloor(b+s)/2\rfloor+1$'' read  ``$d_3\gets\lceil(b-s)/2\rceil$, $d_4\gets\lfloor(b+s)/2\rfloor$''
p. 434\quad line 11, instead of ``$\al/\al'$'' read ``$(\al/\al')\Z_K$''
p. 434\quad line -13, instead of ``whose prime factors are only prime ideals above $2$'' read ``dividing $2$''
p. 439\quad line 2 of (4), instead of ``they will'' read ``it will''
p. 439\quad line 5 of (4), instead of ``repeat'' read ``recompute''
p. 439\quad middle, instead of ``the case of prime degree'' read ``the prime degree case''
p. 444\quad line -8, instead of ``$G_{18}^+$, $G_{18}^-$, $G_{36}$, and $G_{72}$'' read ``$G_{36}^+$, $G_{36}^-$, and $G_{72}$''
p. 451\quad line -3, instead of ``Schwartz's'' read ``Cauchy--Schwarz's''
p. 455\quad line 1, instead of ``Schwartz's'' read ``Cauchy--Schwarz's''
p. 469\quad replace Exercise 6 b) by the following: ``Let $R_1$ denote an even integer such that $0\le R_1\le 2r_1$. Show that the number of $K$-isomorphism classes of quadratic extensions $L/K$ with $\N_{K/\Q}(\gd(L/K))\le x$ and $R_1$ real embeddings is asymptotic to $(\binom{r_1}{R_1/2}/2^{r_1})Q_K\cdot x$, where $Q_K$ is as above.''
p. 470\quad line 1 of Exercise 14, replace `` and'' by ``, ''
p. 471\quad Exercise 16 a), instead of ``Schwartz's'' read ``Cauchy--Schwarz's''
p. 471\quad Exercise 16 a), instead of ``show that'' read ``prove {\it Lagrange's identity\/}''
p. 483\quad line -6, instead of ``$\th_k(\tau)(\th_0(\sigma)^k-1)=0$'' read ``$(\th_0(\sigma)^k-1)\th_k(\tau)=0$''
p. 484\quad line -12, instead of ``$\Gd(L/K)=\Z_L^*$'' read ``$\Z_L^*=\Gd(L/K)^{-1}$''
p. 488\quad middle, instead of ``$v_{\GP}(\f_2)$ is even'' read ``$v_{\GP}(\f_2)$ is even when $\p$ is ramified in $K_2/K$''
p. 520\quad 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 $\al\in L$ be a root of $P$. Assume that the $\p$-adic valuations of all the coefficients of $P$ are nonnegative. Show that $v_{\GP}(\al)\ge0$ for any prime ideal $\GP$ of $L$ above $\p$.''
p. 520\quad line 1 of Exercise 15, instead of ``let $\ell$ is a prime number'' read ``$\ell$ a prime number''
p. 558\quad add at the appropriate place ``$\Gamma_\infty$: group of integer translations, 122''
p. 573\quad add at the appropriate place ``Lagrange's identity, 471''
}
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