\documentclass{article}
\usepackage{amsmath,amssymb,amscd}

\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\N}{\mathbb N}
\newcommand{\F}{{\mathbb F}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\fa}{{\mathfrak a}}
\newcommand{\fc}{{\mathfrak c}}
\newcommand{\fp}{{\mathfrak p}}
\newcommand{\fP}{{\mathfrak P}}
\newcommand{\eps}{\varepsilon}
\newcommand{\lra}{\longrightarrow}
\newcommand{\rsp}{\raisebox{0em}[2.7ex][1.3ex]{\rule{0em}{2ex} }}
\newcommand{\relone}{\mathrel{1}}
\newcommand{\reli}{\mathrel{\hspace{-0.07em}\mbox{\rm l}}}
\newcommand{\smbl}{\relone\joinrel\reli}
\newcommand{\1}{{{}\!\mathrel{\smbl}}}
\newcommand{\Aut}{\mbox{\rm Aut}\,}
\newcommand{\mC}{{\mathcal C}}
\newcommand{\Cl}{\mbox{\rm Cl}\,}
\newcommand{\disc}{\operatorname{disc}\,}

\begin{document}
\section*{Errata and Additions to 
``Reciprocity Laws. From Euler to Eisenstein''.}

%\bigskip

%\subsection*{Minor Corrections}

%\medskip

\begin{tabular}{|l|c|c|}\hline
        \qquad page & for & read \\ \hline
\rsp p. ix, line -- 7 & finite extension & finite normal extension \\ %R. Auer
\rsp p. x, line --12 & that to look & that one should look \\
\rsp p. xi, line --7 & solutions $ax^4 - by^4 = 1$ 
                    &  solutions of $ax^4 - by^4 = 1$ \\ % R. Auer
\rsp p. xiii, line --5 & presentation & presentation of \\ % B. Tangedal
\rsp p. xiv, line 5 & but it hope & but I hope \\  % B. Tangedal
\rsp p. 15, line -- 13   & towards of & towards \\ % R. Auer
\rsp p. 25, line $22$ & $p \equiv q \equiv 1 \bmod 4$ &
                $p \equiv q \equiv 3 \bmod 4$ \\  % Jim Long
\rsp p. 28, Exer. 1.8. & $l_p = rx + sy = tz$ &
                $l_p = rx + sy + tz$ \\  % Jim Long
\rsp p. 30, line 5 & residcovered & rediscovered \\ % Kaplansky
\rsp p. 32, line --  5 & $A, B, M, N \in \N$ & $A, B, M, N \in \Z$ \\
\rsp p. 43, line -- 13 & $X + (X^2-m)\Z[X]$
                       & $X + (X^2-m)\Q[X]$ \\ % B. Tangedal
\rsp p. 44, lines 9--11  & $m < -4 $, $m = -4$ 
              & $\disc k < -4$, $\disc k = -4$ \\ % R. Auer
\rsp p. 47, Proof Prop. 2.8
        & $\lambda^{-1}\fa = \fc^{\sigma - 1}$
        & $\lambda^{-1}\fa = \fc^{1 - \sigma}$ \\
\rsp p. 60, line -- 14  & $p$-adic square  & $2$-adic square \\ % R. Auer
\rsp p. 61, line --3    & implies that & implies \\ % R. Auer
\rsp p. 64, iii) & $\big(\frac{m,n}p\big)$ &
                 $\big(\frac{a,b}p\big)$ \\ % Jim Long, R. Auer
\rsp p. 66, Thm. 2.30 & $\prod_{p \ge 2}$ & $\bigoplus_{p \ge 2}$ \\ % G. Gras
\rsp p. 70, line -- 11 & reciprocity law) & reciprocity law \\
\rsp p. 71, line --  5 & $(p/q) = +1$ & $(p/q) = -1$ \\  % R. Chapman
\rsp p. 73, line --  3 & every factor
                      & every odd factor  \\ % R. Chapman
\rsp p. 74, Ex. 2.29. & $\eps_p$ & $\eps_q$ \\
\rsp p. 76, Ex. 2.29. & 
          $p^h = 4a^2+qb^2$ & $p^h = 4a^2+qb^2$ \\ % G. Cornelissen 
\rsp p. 83, Prop. 3.4 & $\sqrt{p^*}\Z$  & $\sqrt{p^*}\Z[X]$ \\ % R. Auer
\rsp p. 88, Cor. 3.10.vi) & $\fP$ & $\fp$ \\ % R. Auer 
\rsp p. 92, line -- 11  & possible improve & possible to improve \\ % R. Auer
\rsp p. 92, Thm. 3.18 & integers smallest & smallest \\ % R. Auer
\rsp p. 99, lines $8$, $11$ & $\big(\frac{2}{p}\big) \prod$ &
                $\prod$ \\    % P. Roquette
\hline
\end{tabular}

\eject

\begin{tabular}{|l|c|c|}\hline
        \qquad page & for & read \\ \hline
\rsp p. 101, line 6  & $g$ of $p$  & $g$ modulo $p$ \\ % R. Auer
\rsp p. 101, line 9          & $b$ & $a+1$   \\ % R. Auer
\rsp p. 102, footnote line 1 
             & restricted us & restricted ourselves \\ % R. Auer
\rsp p. 111, line -- 13 
             & an $n$-th root & a primitive $n$-th root  \\ % R. Auer
\rsp p. 112, 4.2.iii) & $\alpha \in \cO_k$
                      & $\alpha \in \cO_K$ \\ % B. Tangedal
\rsp p. 112, line 7  & $\xi \in \cO_k$ &  
                $\xi \in \cO_k \setminus \fp$  \\ % R. Auer
\rsp p. 113, line -- 2  & $(\frac{\alpha}{\fp})_k$ 
                & $(\frac{\alpha}{\fp})_K$ \\  % R. Auer
\rsp p. 113, line -- 2   & $(\frac{\alpha}{\fP})_k^{(K:k)}$ 
                & $(\frac{\alpha}{\fP})_K^{(K:k)}$ \\  % R. Auer
\rsp p. 126, line $-3$ & $\pi$ & $\Pi$  \\ % R. Auer
\rsp p. 130, Prop. 4.25 & of $\fp$. 
                       & of $\fp$ in $\Q(\zeta_{mp})$.  \\  % R. Auer
\rsp p. 130, Prop. 4.25.ii) & $\Q(\zeta_m)$ & $\Q(\zeta_{mp})$ \\ 
\rsp p. 130, line 18        & of the Artin & of the Artin symbol  \\  % R. Auer
\rsp p. 131, Prop. 4.28 & $=fp$ & $\fp$  \\  % R. Auer
\rsp p. 135, lines --13, --14 & $\F\1$ & $\cF\1$ \\
\rsp p. 136, line 10 &  [Wy1,Wy2,Why]
                     & [Why] and [Wy1,Wy2] in Ch. 1 \\  % B. Tangedal
\rsp p. 158, line 9 & desired equality (5.5)
                    & desired equality (5.3) \\ %R. Chapman
\rsp p. 162, line -- 2 & $(\frac{p}{q})_4 (\frac{\eps}{p})$ 
                     & $(\frac{p}{q})_4 (\frac{\eps}{q})$ \\
\rsp p. 167, line -- 9 &  $p \mid ABC$  &  $q \mid ABC$ \\
\rsp p. 167, line -- 6 & $m = q$ & $m = p$ \\
\rsp p. 174, line -- 10 & of Chapter 6.7,9 & of Chapters 6, 7 and 9 \\
\rsp p. 181, 5.35 & mod $p$.  & mod $p$, where $p = c^2 + 5d^2$. \\
\rsp p. 189, line $-1$
        & $(\frac{\ell^*}{p})_4$
        & $\big(\frac{\ell^*}{p}\big)_4$ \\
\rsp p. 190, table & 11 & -11 \\
\rsp p. 236, line $-5$ & $a_i \in \Z$ & $a_i \in \Z[\frac12]$ \\
\rsp p. 246, line $-8$ &
         $\phi\Big(\frac{\alpha}{\mu})$
       & $\phi\Big(\frac{\alpha}{\mu}\Big)$ \\ %R. Chapman
\rsp p. 265, line 3 & $K(j(\sqrt{-5})) = K(\sqrt2)$ 
            & $K(j(\sqrt{-5})) = K(\sqrt{-1}\,)$ \\ % R. Auer
\rsp p. 280, Ex. 8.19 &
        $[\kappa/\pi = [\pi^*/\kappa]$
      & $[\kappa/\pi] = [\pi^*/\kappa]$ \\ %R. Chapman
\rsp p. 294, Prop. 9.5.
        & $c \equiv \frac{p-3}4 \bmod 4$
        & $c \equiv -\frac{p+1}4 \bmod 4$ \\ 
\hline
\end{tabular}

\eject

\begin{tabular}{|l|c|c|}\hline
        \qquad page & for & read \\ \hline
\rsp p. 300, lines --8 to -- 6 
             & $\phi(\frac{*}{z})$  & $\phi(\frac{*}{\pi})$ \\ % R. Auer
\rsp p. 302, line $-14$ & did no repeat & did not repeat \\ % B. Tangedal
\rsp p. 312, line 3  & Theorem 9.18 & Theorem 9.19  \\ % R. Auer
\rsp p. 315, Ex. 9.9. & computer. & computer). \\
\rsp p. 318, line 16  & $x^2 + 1 \ne 0$ & $v^2 + 1 \ne 0$  \\ % R. Auer
\rsp p. 319, line 3   &  $(\pm i,0)$, $(0,\pm i)$  
           &  $(w,v) = (\pm i,0)$, $(0,\pm i)$  \\ % R. Auer
\rsp p. 321, line -- 15    & $m$-the & $m$-th \\ % R. Auer
\rsp p. 330, line 3 & $2n \le 50$ & $2n \le 32$ \\ 
\rsp p. 351, Cartier & fonction zeta, & fonction zeta), \\
\rsp p. 355, line -- 4 & J.F. Felipe &  J.F. Voloch \\ %R. Chapman
\rsp p. 372, line 12 & elementattached & element attached \\ %R. Chapman
\rsp p. 372, line 15 & $\sum \alpha<m/2 \sigma_a^{-1}$ &
            $\sum_{\alpha<m/2} \sigma_a^{-1}$ \\ %R. Chapman
\rsp p. 376, lines $-3$, $-1$ & $\eps_i$ & $e_i$ \\
\rsp p. 377, line $2$ & $\eps_i$ & $e_i$ \\
\rsp p. 377, lines $4$, $8$ & $e_\chi$ & $e_i$ \\
\rsp p. 378, lines $2$, $4$, $8$ & $mC_i$ & ${\mathcal C}_i$ \\
\rsp p. 380, line $-12$ & $e_\chi \theta = B_{1,\chi^{-1}} \theta$ &
        $\theta \eps_\chi = B_{1,\chi^{-1}} \eps_\chi$ \\
\rsp p. 394, line --10  & annihilate & annihilates \\
\rsp p. 406, Kleboth & Gle-ichung & Glei-chung \\
\rsp p. 415, Teege 2 & 1921 & 1925 \\
\rsp p. 418, 7th problem & Eisenstein sums & elliptic Gauss sums \\
\rsp p. 422, [55] & Minkowsi--Hasse & Minkowski--Hasse \\
\rsp p. 444, [424] & Yamamoto & Yamamoto, \\
\rsp p. 462, [735] & Sierpinsky & Sierpinski \\  % Kaplansky
\rsp p. 467, [808] & 1895/86 & 1895/96 \\
\hline
\end{tabular}

\vfill \eject

\subsection*{Prop. 1.5.}
This is of course nonsense. What I (probably) meant is that if
$fx^2 + gy^2 = hz^2$ has integral solutions, then $gh$ ($hf$, $-fg$)
are quadratic residues modulo every prime divisor of $f$ ($g$, $h$). %R. Chapman

\subsection*{Lemma 3.13}

The discussion involving the index $m$ ended up in the wrong part of the
proof. Here's the correction:

Assume that there exists a ring homomorphism $\psi: \cO \lra \Z/n\Z$;
since $\Z[\alpha] \subseteq \cO$ is a subring, the composition of
restriction and the isomorphism $\Z[\alpha] \simeq \Z[X]/(f)$ gives a ring
homomorphism $\widehat{\psi}: \Z[X]/(f) \lra \Z/n\Z$. Now let $a \in \N$
be an integer such that $a + n\Z = \widehat{\psi}(X + (f))$. Then
$f(a) + n\Z = \widehat{\psi}(f(X) + (f)) = \widehat{\psi}(0) = 0 + n\Z$,
and so $n \mid f(a)$ as claimed.

Conversely, given $a \in \Z$ with $n \mid f(a)$, we can define a ring
homomorphism $\widehat{\psi}: \Z[X]/(f) \lra \Z/n\Z$ by mapping
$X + (f)$ to $a + n\Z$. This can  be extended to a ring homomorphism
$\cO \lra \Z/n\Z$ by putting $\psi(\beta) = m^{-1}\widehat{\psi}(m\beta)$
for any $\beta \in \cO$: in fact, $m$ is invertible modulo $n$ by assumption,
and $m \beta \in \Z[\alpha]$ for any $\alpha \in \cO$. %R. Chapman


\subsection*{Rational Reciprocity}

Chapter 5, p. 165, line --1: for the last equivalence to be valid
for primes $q \mid C$, we have to interpret the quadratic residue
symbol as a Kronecker symbol (see the bottom of p. 44). Thus in
this case, identity (5.10) shows that
$$ \Big(\frac{A+B\sqrt{m}}{q}\Big) = \Big(\frac2{q}\Big)
     \Big(\frac{A+C\sqrt{m}}{q}\Big),$$
where the symbols on the right hand side are the usual Legendre
symbols.




\subsection*{Herbrand's Theorem}

My ``proof'' of Herbrand's Theorem in Chapter 11 is nonsense.
The confusion arose because I mixed two possible descriptions
of the theorem:
\begin{enumerate}
\item You can look at the quotient group $\mC = \Cl(K)/\Cl(K)^p$
      (as I did) or the subgroup $\Cl(K)[p]$ of ideal classes killed
      by $p$; these are $\F_p[G]$-modules.
\item You can consider the whole $p$-class group $\Cl_p(K)$; in fact, any
      abelian $p$-group $M$ of order $p^{m+1}$ is a $\Z_p[G]$-module in a
      natural way: if $\alpha = a_0 + a_1p + \ldots + a_mp^m + \ldots \in \Z_p$
      (with $0 \le a_j \le p-1$), then putting
      $m^\alpha = m^{a_0 + a_1p + \ldots + a_mp^m}$ for any $m \in M$
      makes $M$ into a $\Z_p[G]$-module.
\end{enumerate}
Both interpretations are compatible: the action of a group ring $RG$
on an abelian $p$-group $M$ gives rise to a homomorphism $RG \lra \Aut(M)$,
and we have a commutative diagram
$$ \begin{CD}
  \Z_pG @>>> \Aut(M) \\
  @VV{\pi}V     @VVV \\
   \F_pG @>>> \Aut(M/M^p)
\end{CD}$$
with $\pi: \Z_pG \lra \F_pG$ being induced by reduction modulo $p$.

We also have to think about the role of the character $\omega$: when
working in $\F_pG$ we simply take $\omega(\sigma_a) = a + p\Z \in \F_p$
and identify $\F_p^\times$ with the roots of unity  $\mu_{p-1}$;
in $\Z_pG$, we have to lift $\omega$ to the Teichm\"uller character
either with Hensel's Lemma or by observing that the sequence
$a$, $a^p$, $a^{p^2}$, \ldots converges in $\Z_p$; its limit
$\omega(a)$ satifies $\omega(a) \equiv a \bmod p$ and
$\omega(a)^{p-1} = 1$, that is: it is contained in the
subgroup $\mu_{p-1} \subset \Z_p^\times$.

Now the proof of Herbrand's Theorem can be done in either way,
but what I did was trying to have the cake and eat it: avoiding
the $p$-adic view and at the same time using the Stickelberger element
$\theta$. This cannot possibly work because $\theta$ has a $p$
in the denominator which is not much of a nuisance in $\Z_p$
(you simply switch to the quotient field) but is of course a
catastrophe in $\F_p$.

Thus in order to save the ``proof'' given in Chapter 11 one has to

\begin{center}
\begin{tabular}{|c|c|}\hline
\rsp  replace  & by \\ \hline
\rsp  $\mC = \Cl(K)/\Cl(K)^p$    & $\Cl_p(K)$ \\
\rsp  $\F_p$   & $\Z_p$ \\
\rsp  cyclotomic $\omega$ & Teichm\"uller $\omega$ \\ \hline
\end{tabular}
\end{center}

\noindent
The corrected proof will be put on my page along with the
rest of Chapter 11.

\vskip 2cm

\subsection*{Additions}

Page 20
Teege's first attempt at filling the gap in Legendre's proof
was incomplete: see his correction in {\em Richtigstellung eines
fr\"uheren Beweises f\"ur den Satz, da\ss\ es f\"ur jede Primzahl
$p$ von der Form $4n+1$ unendlich viele Primzahlen von der Form
$4n+3$ gibt, von denen $p$ quadratischer Nichtrest ist und
Herleitung des Satzes, da\ss\ mindestens eine unter ihnen kleiner
als $p$ ist}, Hamb. Mitt. {\bf 6} (1924), 100--106. In this
paper, Teege also shows that the existence of Gauss's auxiliary
prime follows from Legendre's Lemma.

\medskip

\noindent Page 110, Reference [Te1]: The title of Teege's dissertation is
{\em \"Uber die $\frac{p-1}2$-glie\-dri\-gen Gaussischen Perioden
  in der Lehre von der Kreisteilung und ihre Beziehungen zu
  anderen Teilen der h\"oheren Arithmetik}.

\medskip

\noindent Page 138: the Davenport-Hasse theorem for Jacobi sums (Cor. 4.33)
is actually due to H.H.~Mitchell [{\em On the congruence
$cx^\lambda+1\equiv dy^\lambda$ in a Galois field}, Ann. Math. (2)
{\bf 18} (1917), 120--131], who expressed the result in terms
of cyclotomic numbers.

\medskip

\noindent Page 141: Schwering [728] discusses the quintic power residue
character of $2$, $3$ and $5$ as well as the quintic period
equation.

\medskip

\noindent Page 170: Another proof of the quadratic reciprocity law
in $\Z[i]$ based on Hilbert's genus theory can be found in
S.~Kuroda [{\em \"Uber den Dirichletschen K\"orper},
J. Fac. Sci. Univ. Tokyo, Sect. I {\bf 4} (1943), 383--406].

\medskip

\noindent Page 173: Estes \& Pall [209] give another proof of
Burde's reciprocity law.

\medskip
\noindent Page 200: The version of the quartic reciprocity law
(cf. Exercise 6.17) credited to unpublished papers of Gauss and
Artin already occurs in Busche [107].

\medskip

Appendix B: List of Proofs

J. Sochocki, {\em Bestimmung der constanten Factoren in den Formeln
f\"ur die lineare Transformation der Thetafunctionen. Die Gauss'schen 
Summen und das Reciprocit\"atsgesetz der Legendre'schen Symbole}, 
Par. Denkschrift, 1878, 
gives another proof of the quadratic reciprocity law using theta functions.

B. Tangedal gave a proof of the quadratic reciprocity law for 
Jacobi symbols based on Eisenstein's original proof.

R. Chapman solved the problem of generalizing Nakash's proof 
that $(5/p) = (p/5)$,
giving yet another proof of the quadratic reciprocity law.

Two more proofs were given by \ldots

% Page 336: Whether Gauss and Herglotz counted the infinite solutions
%  twice or not is not so clear: they might as well have realized
%  that there are four infinite solutions if the singularity at
%  $\infty$ is resolved. This is, however, pure speculation.

\end{document}
