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\begin{document}

\begin{center}
\textbf{Algebra: Abstract and Concrete}

\textit{Frederick M. Goodman}

Edition 2.6 (last revised May 1, 2015)

\url{https://homepage.divms.uiowa.edu/~goodman/algebrabook.dir/download.htm}

\textbf{Errata and comments} by Darij Grinberg
\end{center}

\noindent
The page numbers below refer to the printed page numbers in Edition 2.6.
This list is not claimed to be exhaustive.  Items marked ``substantive''
change a mathematical assertion or supply a missing hypothesis; other items
are local corrections or clarifications.  Formula-sensitive items have been
checked against rendered pages of the PDF, since text extraction from this
file occasionally loses minus signs.

The majority of errata in this list were found and written up by GPT-5.6 Sol.
All have been verified by Darij Grinberg.

\appendix
\setcounter{section}{7}

\section{Corrections and mathematical clarifications}

\begin{enumerate}%[leftmargin=2em]

\item \textbf{Page 22, Theorem 1.5.3 (uniqueness of disjoint-cycle
notation):} \emph{Clarification.}
This theorem, as stated, relies on the convention that each cycle
has length $\geq 2$ (that is, it permutes at least two elements).
If $1$-cycles (i.e., cycles of length $1$) were allowed, then the
representation of a permutation as a product of disjoint cycles
would no longer be unique, since $1$-cycles are just the identity
permutation (after all, cycling through
a single element obviously leaves this element fixed!) and thus
can be inserted into a product at will without affecting the
product.

Later on -- e.g., in Example 5.1.17 and Exercise 5.1.17 -- this
convention is abandoned; there, you allow cycles of length $1$,
and in fact require them in certain cases.
Namely, in Chapter 5, you include a $1$-cycle $\left(i\right)$
for each fixed point $i$ of the permutation, so as to ensure that
each element of $\left\{1,2,\ldots,n\right\}$ appears exactly 
once in the cycle notation.
For example, the permutation in $S_9$ that would be written as
$\left[\left(2 \ 5\right) \left(4 \ 6 \ 9 \ 8\right)\right]$
according to the convention of Chapter 1 would be written as
$\left[\left(1\right) \left(2 \ 5\right) \left(3\right)
\left(4 \ 6 \ 9 \ 8\right) \left(7\right) \right]$ in Chapter 5
(with the redundant $1$-cycles being included in order to ensure
that each element of $\left\{1,2,\ldots,n\right\}$ appears exactly
once).
With this convention, the representation is again unique.

\item \textbf{Page 63:} It is worth mentioning that $1$ means
the function $\mathbf{1}_U : U \to \{0,1\}$,
which sends every $u \in U$ to $1$.

\item \textbf{Page 153, Example 3.1.11:}
There are three Dalmatian dogs, so their permutation group $D$ is
isomorphic to $S_3$, not to $S_4$.

Moreover, the displayed copy of $A$ inside
\[
  P=A\times B\times C\times D\times E
\]
should be
\[
  \widetilde A
  =A\times\{e\}\times\{e\}\times\{e\}\times\{e\},
\]
with four trivial factors $\times\{e\}$, not three.

\item \textbf{Page 155, Definition 3.1.16:}
In both formulas defining the coordinatewise operations, replace
the final entry $r_s'$ of the second tuple by $r_n'$.

\item \textbf{Page 155, Proposition 3.1.18:}
``For any integers $x_1,x_2,\ldots,x_s$'' should be
``For any integers $x_1,x_2,\ldots,x_n$''.

\item \textbf{Page 162, Exercise 3.2.2:}
The automorphism
\[
  j:[x]\longmapsto[-x]
\]
does not always have order $2$: it is the identity for $n=1$ and
$n=2$.  It does satisfy $j^2=\id$ for every $n$, which is all that
is needed to define the indicated homomorphism from $\ZZ_2$.
Thus, either replace ``an order $2$ automorphism'' by
``an automorphism satisfying $j^2=\id$'', or add the hypothesis
$n>2$ to the assertion about its order.

\item \textbf{Page 172, proof of Proposition 3.3.25:}
``there is a nonzero
$\boldsymbol{\alpha}=\begin{bmatrix}\alpha_1\\ \vdots \\ \alpha_n
\end{bmatrix} \in K^n$''
should be
``there is a nonzero
$\boldsymbol{\alpha}=\begin{bmatrix}\alpha_1\\ \vdots \\ \alpha_s
\end{bmatrix} \in K^s$''.

\item \textbf{Page 172, proof of Corollary 3.3.26:}
The second application of Proposition 3.3.25, with the roles of $X$ and $Y$
reversed, gives $|X|\leq |Y|$, not a second copy of $|Y|\leq |X|$.

\item \textbf{Page 175, proof of Proposition 3.3.33:}
The proof cites Proposition 3.5.1, which has not yet appeared.
It would be natural to switch the order of these two results.

\item \textbf{Page 176, Proposition 3.3.35:}
The preceding argument assumes that $T:V\to W$ is surjective, whereas
the proposition is stated for an arbitrary linear map.  To obtain the
stated result, apply the preceding argument to the surjective map
$T:V\longrightarrow\operatorname{range}(T)$.

\item \textbf{Page 176, warning after Corollary 3.3.37:}
The word ``never'' in ``Complements of a subspace are never unique''
is an overstatement:
For example, the zero subspace has the unique complement $V$, and
$V$ has the unique complement $\{0\}$.  Replace ``are never unique''
by ``need not be unique'' or ``are usually not unique''.

\item \textbf{Page 177, Exercise 3.3.10 (a):}
The row $  [v_1,\ldots,v_n]$
belongs to $V^n$, not to $K^n$.

\item \textbf{Page 182, paragraph after Theorem 3.4.5:}
The assertion that the canonical map $V\to V^{**}$ is not surjective
when $V$ is infinite-dimensional is true, but is nontrivial and is
neither proved nor assigned as an exercise.

\item \textbf{Pages 184--185, Proposition 3.4.10:}
The convention introduced immediately before the proposition denotes
the matrix of $T:V\to W$ by $[T]_{C,B}$, where $B$ is the basis of
$V$ and $C$ the basis of $W$.  Thus, in part (a), replace
``$T\longmapsto[T]_{B,C}$'' by ``$T\longmapsto[T]_{C,B}$''.
Make the same correction in the final sentence of the proof of
part (a).

\item \textbf{Page 188, Exercise 3.4.4:}
In the displayed inner product, the second vector should be
\[
  \begin{bmatrix}\beta_1\\ \beta_2\\ \beta_3\end{bmatrix},
\]
not
\[
  \begin{bmatrix}\beta_2\\ \beta_2\\ \beta_3\end{bmatrix}.
\]

\item \textbf{Page 189, Exercise 3.4.10 (b):}
The ``integration'' map $P_6\to P_7$ is not determined until a choice
of integration constant has been specified.  For example, one can
define it to be
$f(x)\longmapsto\int_0^x f(t)\,dt$,
equivalently the unique antiderivative with constant term $0$.

\item \textbf{Page 190, first paragraph of Section 3.5:}
After ``the order of an element $x$ is the smallest
natural number $s$ such that $sx=0$'', add ``(or $\infty$ if
no such $s$ exists)''.

\item \textbf{Page 194, Proposition 3.5.9:}
The integer $s$ in
\[
  \operatorname{diag}(d_1,\ldots,d_s,0,\ldots,0)
\]
is not introduced.  One should specify that
$0\leq s\leq\min\{m,n\}$ and that $d_1,\ldots,d_s$ are the nonzero
diagonal entries.

\item \textbf{Page 200, derivation of the invariant-factor decomposition:}
In the direct product
\[
  \ZZ/d_i\ZZ\times\cdots\times\ZZ/d_s\ZZ\times\ZZ^{\,n-s},
\]
the first factor should be $\ZZ/d_1\ZZ$, not $\ZZ/d_i\ZZ$.

\item \textbf{Page 203, proof of uniqueness in Theorem 3.6.2:}
The proof should dispose separately of the case
$G_{\mathrm{tor}}=\{0\}$ before writing
\[
  a_s=b_t=a.
\]
In that case $s=t=0$, so neither $a_s$ nor $b_t$ exists, and uniqueness
is immediate.  Add, for example:
``If $G_{\mathrm{tor}}=\{0\}$, then $s=t=0$, and there is nothing
left to prove.  Hence assume $G_{\mathrm{tor}}\neq\{0\}$.''

\item \textbf{Page 204, proof of uniqueness in Theorem 3.6.2:}
After ``Let $k'$ be the last index such that $a_{k'}=p$'', add ``(or $0$
if no such index exists)''.
Make a similar change when $k''$ is defined.

\item \textbf{Page 204, Example 3.6.6:}
Under the convention in Theorem 3.6.2 that the invariant factors satisfy
$a_1\mid a_2\mid\cdots$, the final decomposition should be written
\[
  \ZZ_{30}\times\ZZ_{24}\cong \ZZ_6\times\ZZ_{120},
\]
and the invariant factors should be listed as $6,120$, rather than $120,6$.
(The two displayed direct products are of course isomorphic; the issue is the
ordering convention.)

\item \textbf{Page 205, Definition 3.6.9 (b):}
If the partition is denoted $(n_1,n_2,\ldots,n_k)$, then the displayed
decomposition should end in $\ZZ_{p^{n_k}}$, not in
$\ZZ_{p^{n_s}}$.

\item \textbf{Page 207, proofs of Theorem 3.6.15 and
Corollary 3.6.16:}
In both prime factorizations, replace
\[
  p_1^{k_1}p_s^{k_2}\cdots p_s^{k_s}
\]
by
\[
  p_1^{k_1}p_2^{k_2}\cdots p_s^{k_s}.
\]

\item \textbf{Pages 210--212, Examples 3.6.22--3.6.24:}
The invariant factors are systematically listed in the reverse of the
order stipulated in Theorem 3.6.2, where
$a_1\mid a_2\mid\cdots\mid a_s$.

For instance, the decomposition on page 210 should be written
\[
  \ZZ_2\times\ZZ_{10}\times\ZZ_{2100},
\]
rather than
\[
  \ZZ_{2100}\times\ZZ_{10}\times\ZZ_2.
\]
Likewise, the algorithm on pages 210--211 produces the invariant
factors from largest to smallest, so the resulting list should be
reversed before it is put into the convention of Theorem 3.6.2.

On page 212, the six decompositions should accordingly be written as
\[
\begin{gathered}
  \ZZ_{4200},\\
  \ZZ_5\times\ZZ_{840},\\
  \ZZ_2\times\ZZ_{2100},\\
  \ZZ_{10}\times\ZZ_{420},\\
  \ZZ_2\times\ZZ_2\times\ZZ_{1050},\\
  \ZZ_2\times\ZZ_{10}\times\ZZ_{210}.
\end{gathered}
\]
The displayed groups are of course unchanged up to isomorphism; the
problem is only the stated ordering convention.

\item \textbf{Page 213, proof of Proposition 3.6.27 (b):}
Exercise 2.2.30 correctly shows that the order of $[3]$ in
$\Phi(2^n)$ is $2^{n-2}$, not $2^{n-1}$.
Consequently, replace ``order $2^{n-1}$'' by ``order $2^{n-2}$'',
and replace each of the three ``$\ZZ_{2^{n-1}}$''s
by ``$\ZZ_{2^{n-2}}$.

\item \textbf{Page 215, Exercise 3.6.17:}
If $(m_1,\ldots,m_k)$ are invariant factors with
$m_1\mid m_2\mid\cdots\mid m_k$, then a finite abelian group $G$ has an
element of order $s$ if and only if $s\mid m_k$, not if and only if
$s\mid m_1$.  The exponent of $G$ is the largest invariant factor $m_k$.

\item \textbf{Page 249, Exercise 5.1.20:}
The proposed description
\[
  \ZZ_4=\left\langle (1\ 2\ 3\ 4),(1\ 2)(3\ 4)\right\rangle
\]
is wrong: the displayed generators generate the dihedral group
$D_4$, exactly as in the preceding bullet point.  Replace this by
\[
  \ZZ_4=\left\langle (1\ 2\ 3\ 4)\right\rangle .
\]

\item \textbf{Page 250, proof of Proposition 5.2.2:}
The characteristic function $\mathbf 1_F$ was defined on
$G\times X$, not on $X \times G$.
Thus, in both displayed double sums, replace
$\mathbf 1_F(x,g)$ by $\mathbf 1_F(g,x)$.

\item \textbf{Page 259, paragraph following the proof of
Theorem 5.4.11:}
In the summary of the Sylow theorems, replace
``the number of such subgroups divides $|G|$ and is conjugate to
$1$ mod $p$'' by
\[
  \text{``the number of such subgroups divides $|G|/p^n$
  and is congruent to $1$ modulo $p$.''}
\]
The statement that it divides $|G|$ is true but weaker
than the third Sylow theorem
(although easily seen to be equivalent in view of the
$\equiv 1 \mod p$ property),
whereas the word ``conjugate'' is simply wrong.

\item \textbf{Page 260, Example 5.4.14:}
``Hence $n_r\in\{1,10\}$'' should be
``Hence $n_3\in\{1,10\}$''.

\item \textbf{Page 261, Example 5.4.14:}
The assertion that the map
\[
  [k]\longmapsto[-k]
\]
is an automorphism of order $2$ of $\ZZ_n$ ``for any $n$'' is false
for $n=1$ and $n=2$; in these cases it is the identity.  Replace
``for any $n$'' by ``for any $n>2$''.  Only $n=3,5$ is needed here.

\item \textbf{Page 263, Exercise 5.4.10, converse assertion:}
The relations
\[
  a^7=b^4=1,\qquad bab^{-1}=a^{-1}
\]
do not by themselves force a group generated by $a,b$ to be the indicated
semidirect product; they only make it a quotient of that group.  For example,
one may take $a=1$.  Add, for instance, that $a$ and $b$ have orders exactly
$7$ and $4$, respectively (or that the generated group has order $28$), or
rephrase the assertion as a presentation of the semidirect product.

\item \textbf{Page 264, first non-italicized paragraph of Section 5.5:}
The subgroup
\[
  \left\langle(1\ 2\ 3\ 4\ 5),(1\ 2)\right\rangle
\]
is $S_5$, not $D_5$.  For example, conjugating
$(1\ 2)$ by powers of $(1\ 2\ 3\ 4\ 5)$ produces enough
transpositions to generate $S_5$.
Replace $(1\ 2)$ by, for example,
\[
  (2\ 5)(3\ 4),
\]
as in Proposition 5.5.3.

\item \textbf{Page 265, paragraph immediately before
Proposition 5.5.3:}
Since the common normalizer of the three displayed subgroups has
order $20$, its index in $S_5$ is
\[
  [S_5:\langle\sigma,\rho\rangle]=120/20=6,
\]
not $5$.  Thus replace ``the number of conjugates of each of the
groups is $5$'' by ``\ldots is $6$''.
In Proposition 5.5.3, ``and its five conjugates'' is consistent with
this if it means five \emph{other} conjugates; ``and its five other
conjugates'' would be clearer.

\item \textbf{Page 266, Definition 5.6.1:}
Replace ``if for all $x\in X$,''
by ``if for all $g\in G$ and all $x\in X$, we have''.

\item \textbf{Page 267, Exercise 5.6.6:}
Add the hypothesis $n\geq 3$.  For $n=2$, the point stabilizer
$H=\operatorname{Stab}(2)$ is the trivial subgroup of $S_2$,
whose normalizer is all of $S_2$, not $H$.

\item \textbf{Page 268, Exercise 5.6.18:}
This exercise is false.  Let
\[
  V=\left\{(x_1,\ldots,x_5)\in\mathbb F_2^5
  \ \middle|\ x_1+\cdots+x_5=0\right\}
  \cong(\ZZ_2)^4,
\]
and let $\ZZ_5$ act nontrivially on $V$ by cyclically permuting the
five coordinates.  Then
\[
  G=V\rtimes\ZZ_5
\]
has order $2^4\cdot5=80$, but the displayed complement $\ZZ_5$ is
not normal.  Indeed, $G$ has multiple $5$-Sylow subgroups
(since any $(v, c) \in V \rtimes \ZZ_5 = G$ for which $c$
is nonzero has order $5$ in $G$), which are all conjugate
by the Second Sylow Theorem, and thus none of them is normal.
% OLD PROOF: if both $V$ and this complement were normal,
% then their commutator would lie in their trivial intersection, so the
% action would be trivial.

A simple repair is to assume $1\leq n\leq3$.  In that case the number
of $5$-Sylow subgroups divides $2^n$ and is congruent to $1$ modulo
$5$, and hence must equal $1$.

\item \textbf{Page 270, etc.:}
It might also be worthwhile to point out that polynomials usually
``know'' the names of their variables; i.e., the polynomial
$x+1 \in K[x]$ is not
identified with $y+1 \in K[y]$ (or else things would go awry when
$K[x]$ and $K[y]$ are embedded into $K[x,y]$). The definition of
polynomials as sequences (or families, in the multivariate case) of
coefficients obscures this point, as it suggests that the monomials
$x^2 y$ in $K[x,y]$ and $y^2 z$ in $K[y,z]$ both are just the
multi-index $(2, 1)$. To avoid this ambiguity, it should be
clarified that a monomial is not \textit{just} the multiindex, but
also the list of names of variables involved -- e.g., the monomial
$x^2 y$ in $K[x,y]$ is the multi-index $(2, 1)$ with the list of
names $(``x'',``y'')$, whereas the monomial 
$y^2 z$ in $K[y,z]$ is the multi-index $(2, 1)$ with the list of
names $(``y'',``z'')$.

\item \textbf{Page 274, Exercise 6.1.12:}
The claim that ``$R$ is isomorphic to the
subring of constant functions on $X$'' is only true if $X$ is nonempty.

\item \textbf{Page 284, Proposition 6.2.27 (a):}
The displayed description of $R\mathcal{S}R$ should specify all three families of
conditions:
\[
 R\mathcal{S}R=\left\{\sum_{i=1}^{n}a_i s_i b_i:\ n \geq 0,\ a_i,b_i\in R,
       \ s_i\in\mathcal{S}\right\}.
\]
As printed, only $a_n,b_n\in R$ are mentioned, and no condition at all is
placed on the $s_i$.
% If the empty subset $S=\varnothing$ is allowed, one
% must also allow the empty sum (or separately declare $R\varnothing R=\{0\}$).

\item \textbf{Page 295, Definition 6.4.2:}
\emph{Missing hypothesis.}
To agree with the standard convention and with subsequent uses, require an
integral domain to be nontrivial, equivalently to satisfy $1\neq 0$.  Otherwise
the zero ring satisfies the printed condition vacuously.

\item \textbf{Page 298, Definition 6.4.10 and Proposition 6.4.11:}
\emph{Missing hypothesis.}
A prime ideal is required to be proper.  Thus Definition 6.4.10 should add
the requirement $J\neq R$.  Correspondingly, Proposition 6.4.11 should say that $J$ is
prime if and only if $R/J$ is a nonzero ring with no zero divisors (and, in
the unital commutative case, an integral domain in the corrected sense).

\item \textbf{Page 298, Example 6.4.13:}
Every ideal of $\ZZ$ has the form $d\ZZ$ for a unique $d\geq 0$, not
necessarily for $d>0$.  With the usual definition of prime ideal, $(0)$ is
also a prime ideal of $\ZZ$.  Thus the positive ideals $d\ZZ$ are prime
exactly when $d$ is a prime number, and $(0)$ is the additional prime ideal.

\item \textbf{Page 300, definition of a Euclidean domain:}
The condition ``$d(fg) \geq \max\{d(f), d(g)\}$'' is not used until
Lemma 8.4.7 (and possibly the solution to Exercise 6.5.3, where it
allows for a simpler proof).
Several authors omit it from the definition of a Euclidean domain.

\item \textbf{Page 317, Lemma 6.7.4 and its proof:}
First dispose separately of the case $J=(0)$, since otherwise the ``minimum
of the degrees of nonzero elements of $J$'' is undefined.  In the induction
step, ``choose $g\in J_0$ such that
$\deg(f-g)<\deg(g)$'' should read
``choose $g\in J_0$ such that
$\deg(f-g)<\deg(f)$''.
% This is both what the hypothesis supplies and what is needed for the
% induction.

\item \textbf{Page 324, display (7.2.1):}
The first elementary-symmetric relation should be
\[
  \alpha_1+\alpha_2+\alpha_3=0,
\]
not $\alpha_1+\alpha_3+\alpha_3=0$.

\item \textbf{Page 324, Cardano formula near the bottom of the page:}
The second root is labeled $a_2$; it should be $\alpha_2$.
More importantly, the claim that the square-root choice is immaterial needs
an exception:  When $p=0$, one of the two square-root choices can make
$A^3=0$, after which the displayed expression $p/(3A)$ is undefined.
Choose the sign so that $A\neq 0$ when possible, and handle the case
$p=q=0$ separately (where the sole root is $0$ with multiplicity $3$).

\item \textbf{Page 356, Definition 8.1.28:}
In my view, this is a bad definition, as it lacks the flexibility
that is later needed (and silently assumed). It makes a lot
more sense to define linear independence for \textit{families} or
\textit{tuples} (rather than \textit{sets}), and similarly to say that
a \textit{family} or a \textit{tuple} (not a \textit{set}) is a basis
of an $R$-module. This does not mean
that it is always wrong to speak of the linear independence of a set
(it is a particular case of the linear independence of a family:
namely, the family $(s)_{s \in S}$, where $S$ is the set in question),
but \textit{most of the time} when people say ``the set
$\{x_1, \ldots, x_n\}$ is linearly independent'', they really mean
``the tuple $(x_1, \ldots, x_n)$ is linearly independent''.
This sort of language often causes statements to mean different things
than the author meant to say. For example, on page 328, in the proof
of Proposition 7.3.4, you say that the powers $1, \alpha, \alpha^2,
\alpha^3, \ldots$ cannot be linearly independent over $K$. You definitely
mean that the family $(\alpha^n)_{n \geq 0}$ is not linearly
independent, rather than the set $\{1, \alpha, \alpha^2, \alpha^3,
\ldots\}$. (The latter would be false for $\alpha = 1$, because in this
case the set $\{1, \alpha, \alpha^2, \alpha^3, \ldots\} = \{1\}$ is
linearly independent.)
On page 384, in Exercise 8.4.1 (b), the set $\{v_1, v_2, \ldots, v_n\}$
does not change
if we duplicate some of the $v_i$'s, but the claim of the exercise
becomes wrong. In Example 8.5.4, you speak of ``the set of $n+1$
elements $\{v, T(v), T^2(v), T^3(v), \ldots, T^n(v)\}$'', but this doesn't
have to be a set of $n+1$ elements; it is a \textit{tuple} of $n+1$
elements.
On page 398, you say that ``the $n^2+1$ linear transformations $\id, T,
T^2, \ldots, T^{n^2}$ are not linearly independent'', and again this only
holds if you take their tuple rather than their set.
(On the other hand, Proposition 9.5.7 is one of the few places where
it really matters that the set is a set. Here I'd replace ``collection''
by ``set'' to stress this.)

\item \textbf{Page 375, Lemma 8.4.1:}
Here you should assume that $R$ is not the
zero ring. Otherwise, the zero ring over itself is a counterexample,
having a basis of any size (as I said, bases are naturally tuples or
families, not sets; but even if you restrict yourself to sets, then
there is still a basis of size $0$ and a basis of size $1$). In
\href{https://doi.org/10.1090/S0002-9939-1988-0954974-5}{his paper
\textit{Nontrivial uses of trivial rings} (Proc. Amer. Math. Soc.
\textbf{103} (1988), pp. 1012--1014)}, Fred Richman makes a highly
convincing case that the ``right'' way to
state the correct lemma is to say that if $R$ is a commutative ring
with unity, and $M$ is an $R$-module equipped with two bases of
distinct finite cardinalities, then $R = 0$.

\item \textbf{Page 387, Example 8.5.3:} Replace
``there is an $n \in \ZZ$'' by ``there is a nonzero
$n \in \ZZ$''.

\item \textbf{Page 448, Definition 9.6.5:}
Algebraic independence means that there is no \emph{nonzero} polynomial
$f\in K[x_1,\ldots,x_n]$ satisfying $f(u_1,\ldots,u_n)=0$.  The zero
polynomial always gives such a relation, so the word ``nonzero'' is
essential.

Also, it is better to define this notion
over rings instead of fields; i.e., let $K$ be a commutative ring with
unity, and let $u_1, \ldots, u_n$ lie in a commutative $K$-algebra with
unity. This way, the first statement in Theorem 9.6.6 applies to every
commutative ring $K$. (Also, again, I am not keen on speaking of
``algebraically independent sets'' as opposed to ``algebraically
independent families''.)

\item \textbf{Page 449:}
Your definition of partitions misses one ingredient: two
partitions that only differ in trailing zeroes should be identified.
For example, $(3, 2, 2) = (3, 2, 2, 0) = (3, 2, 2, 0, 0) = \cdots$.
Otherwise, the conjugate partition $\lambda^\ast$ would not be
uniquely defined (or would fail to satisfy $(\lambda^\ast)^\ast =
\lambda$) due to too much freedom in choosing its length.

\item \textbf{Page 450, definition of the monomial symmetric function
$m_\lambda$:} The formula
\[
  m_\lambda=(1/f)\sum_{\sigma\in S_n}\sigma(x^\lambda)
\]
is not defined over an arbitrary field $K$ when the stabilizer size $f$ is
zero in $K$.  Instead, $m_\lambda$ should be defined as
\[
  m_\lambda=\sum_{u\in S_n\cdot x^\lambda}u,
\]
the sum of the distinct monomials in the orbit of the monomial $x^\lambda$
under the symmetric group $S_n$ permuting the variables.
% (equivalently, sum over a set
% of coset representatives for the stabilizer).

\item \textbf{Pages 450--451, Lemma 9.6.7 (c):}
The inverse coefficients $S_{\lambda\mu}$ need not be nonnegative.  For
example,
\[
  m_{(2)}=\epsilon_1^2-2\epsilon_2.
\]
Replace ``nonnegative integer'' in part (c) by ``integer''.  The proof by
inverting a unitriangular integer matrix establishes integrality, not
positivity.

\item \textbf{Page 450, proof of Lemma 9.6.7:}
``number of $\lambda_k$'' should be ``number of $k$''
(we must count the $k$'s, not the $\lambda_k$'s; for example,
if $\lambda_2 = \lambda_3$, then both $2$ and $3$ are counted).

\item \textbf{Page 453, Exercise 9.6.15:}
This needs a requirement that $\operatorname{char} K \neq 2$.
(Alternatively, retain the condition
$\sigma(f)=\epsilon(\sigma)f$ and additionally require $f$ to
vanish whenever two of its indeterminates are set equal.
With these two conditions, the conclusion holds in every
characteristic.)

\item \textbf{Page 461, beginning of Section 9.8:}
\emph{Missing hypothesis.}
The substitution $x=y-a/4$ and formulas (9.8.3) require at least
$\operatorname{char}K\neq 2$.  This assumption should be made before the
substitution, rather than only later in the discussion.

\item \textbf{Page 461, formula (9.8.2):}
Replace both ``$x$''s on the right-hand side of (9.8.2) by
$y$'s. That is, the formula should be
\[
  f(x)=g(y)=y^4+py^2+qy+r.
\]

\item \textbf{Page 462, Case 1A:}
If the resolvent cubic $h$ is irreducible and its discriminant is a square,
then its splitting field has degree $3$, not $6$.  Indeed its transitive
Galois group is a subgroup of $A_3$, hence is $A_3\cong C_3$.  Thus the argument
should say that $3$ divides $|G|$, which still distinguishes $A_4$ from the
Klein four group and yields $G=A_4$.

\item \textbf{Page 463, Lemma 9.8.1, first two sentences:}
The variables are inconsistent.  Write, for example,
\[
  g(x)=x^4+px^2+qx+r
\]
and let $h(y)$ denote its resolvent cubic (or consistently use $y$ for the
quartic variable throughout), rather than writing $g(x)=y^4+py^2+qy+r$.

\item \textbf{Page 479, alternative proof of simplicity of $A_n$:}
\emph{Substantive terminology and notation.}
The stabilizer of a subgroup $N$ under conjugation is its
\emph{normalizer}, not its centralizer.  Thus ``centralizer'' should
be replaced by ``normalizer'' throughout this page, and
``$\operatorname{Cent}_{S_n}(N)$'' should be likewise replaced by
``$N_{S_n}(N)$''.

\item \textbf{Page 485, Lemma 10.6.3:}
This is false unless the extension $L/K$ is separable.
(The classical example $K = \mathbb{F}_p\left(x^p\right)$ and
$L = \mathbb{F}_p\left(x\right)$ of an inseparable finite extension
is a counterexample.)

\item \textbf{Pages 485--487, proof of Theorem 10.6.2:}
Even aside from the problem with Lemma 10.6.3 noted above, this proof
does not work in arbitrary positive characteristic.  If
$p=\operatorname{char}K$ divides
$n=n_1n_2\cdots n_r$, then no primitive $n$-th root of unity exists in
any extension field of $K$.  Thus the instruction to adjoin such a root
cannot be carried out.
% Parts of the proof do work if a primitive $n$-th root of unity is
% understood in the weaker sense ``an element $u$ of the field such that
% all roots of $x^n - 1$ \textbf{in a splitting field} of the polynomial
% $x^n - 1$ are powers of $u$''.
% (This requires a slight generalization to Lemma 10.5.1 and
% Proposition 10.5.2, because as these two facts are stated, they require
% $n$ to be relatively prime to the characteristic of $K$.)

\item \textbf{Page 492, definition of the Euclidean norm:}
The displayed formula
\[
  \|\boldsymbol{a}\|=\langle \boldsymbol{a},\boldsymbol{a}\rangle=\sum_i a_i^2
\]
 should read
\[
  \|\boldsymbol{a}\|^2=\langle \boldsymbol{a},\boldsymbol{a}\rangle=\sum_i a_i^2,
  \qquad\text{or equivalently}\qquad
  \|\boldsymbol{a}\|=\sqrt{\langle \boldsymbol{a},\boldsymbol{a}\rangle}.
\]

\item \textbf{Page 493, proof of Lemma 11.1.1:}
Both displayed equalities have the wrong sign in front of the inner product.
They should be
\[
  \|\boldsymbol{a}-\boldsymbol{b}\|^2
  =\|\boldsymbol{a}\|^2+\|\boldsymbol{b}\|^2
  -2\langle \boldsymbol{a},\boldsymbol{b}\rangle
\]
and the analogous identity for $\tau(\boldsymbol{a}),\tau(\boldsymbol{b})$.
The desired conclusion still follows after making this correction.

\item \textbf{Page 497, Lemma 11.1.9:}
The formula ``$\sigma \tau_{\boldsymbol{b}} \sigma^{-1} =
\tau_{\sigma(\boldsymbol{b})}$'' should instead be
\[
  \sigma\tau_{\boldsymbol{b}}\sigma^{-1}=\tau_{\boldsymbol{A}\boldsymbol{b}},
  \qquad
  \text{ where } \sigma \text{ is given by }
  \sigma(\boldsymbol{x}) = \boldsymbol{A} \boldsymbol{x} + \boldsymbol{c}.
\]

\item \textbf{Page 501, proof of Theorem 11.2.6:}
The induction begins with graphs having exactly one edge, but the theorem
also includes the connected graph with one vertex and no edges.  Add this
base case: $v=1$, $e=0$, $f=1$, so $v-e+f=2$.

The tree step also invokes
the existence of a vertex of valence $1$, a fact deferred to Exercise
11.2.4; it would be clearer to state this as a preceding lemma.

\item \textbf{Page 502, proof of Euler's theorem for polyhedra:}
If ``polyhedron'' denotes the solid convex body, the radial projection
$\boldsymbol{x}\mapsto \boldsymbol{x}/\|\boldsymbol{x}\|$
does not map the whole polyhedron bijectively onto the
sphere.  It maps the \emph{boundary} of a convex polyhedron containing the
origin bijectively onto the sphere.  Replace ``maps the polyhedron
bijectively'' by ``maps the boundary of the polyhedron bijectively'' (or
state explicitly that ``polyhedron'' here means its boundary surface).

\item \textbf{Page 550:}
The notation $M^j$ for the $j$-th column of a matrix $M$ is rather
confusing, even if the matrix $M$ is (in general) rectangular and thus
has no $j$-th power.

\item \textbf{Page 551, Definition E.9:}
Replace ``$\{T(\boldsymbol{x}):\boldsymbol{x}\in K^n\}$'' by
``$\{T(\boldsymbol{x}):\boldsymbol{x}\in V\}$''.

\end{enumerate}

\section{Minor typographical and editorial corrections}

\begin{enumerate}

\item \textbf{Page 32, paragraph after Proposition 1.6.12:}
``it is a only a short way'' should be ``it is only a short way''.

\item \textbf{Page 55, Exercise 1.8.12:} There is an
extraneous closing parenthesis at the end of this exercise.

\item \textbf{Page 62, Inclusion--Exclusion:}
``but then uncounted once in $\left|A\cap B\right|
- \left|A\cap C\right| - \left|B\cap C\right|$'' should be
``but then uncounted once in $-\left|A\cap B\right|
- \left|A\cap C\right| - \left|B\cap C\right|$''.

\item \textbf{Page 66, computation of $\varphi(n)$:}
Replace ``$A'_1 \cap A'_2 \cap \cdots \cap A'_n$''
by ``$A'_1 \cap A'_2 \cap \cdots \cap A'_s$''.

\item \textbf{Page 76, first paragraph:}
Delete one occurrence of ``with'' in
``or with with polynomial entries''.

\item \textbf{Page 78, Proposition 1.11.7:}
Replace ``$Z_b$'' by ``$\ZZ_b$''.
Make the same change in the proof of the proposition.

\item \textbf{Page 81, proof of Lemma 1.12.2:}
Remove ``Write $rs=1+tm$'', since you never use $t$.
You just use the assumption $rs \equiv 1 \mod m$.

\item \textbf{Page 105, Exercise 2.2.23:}
``therein $\ZZ_{200000}$'' should be ``there in $\ZZ_{200000}$''.

\item \textbf{Page 119, Exercise 2.4.14:} Replace ``$k$ cycle'' and
``$k$ cycles'' by ``$k$-cycle'' and ``$k$-cycles'', respectively.

\item \textbf{Page 157, Exercise 3.1.2:}
``$\{e_a\}$'' should be ``$\{e_A\}$''.

\item \textbf{Page 158, Exercise 3.1.7:}
``Show that $\pi_i$ surjective homomorphism'' should be
``Show that $\pi_i$ is a surjective homomorphism''.

\item \textbf{Page 164:}
``The \emph{kernel} of linear transformation'' should be
``The \emph{kernel} of a linear transformation''.
Also, in Lemma 3.3.3, ``then'' at the beginning of the second sentence
should be capitalized.

\item \textbf{Page 165, quotient-vector-space paragraph:}
Delete the repeated word in ``check that this this is well-defined'', and
replace ``if $v+W=v'+W$ and, then'' by ``if $v+W=v'+W$, then''.

\item \textbf{Page 166:}
``straighforward'' should be ``straightforward'', and
``indclude'' should be ``include''.

\item \textbf{Page 169, discussion of the empty set:}
The statement that the empty set is linearly independent because there
are no sequences of its elements is correct if ``sequence'' means a
sequence of positive length, as in the book's convention.  If sequences
of length $0$ are allowed (which is the standard convention in
mathematics), there is exactly one such sequence, namely
the empty sequence; its coefficient vector is the unique element of
$K^0$, so the conclusion about linear independence remains correct.

\item \textbf{Page 172, Proposition 3.3.25:}
``Let $V$ a finite dimensional vector space'' should be ``Let $V$ be
\ldots''.

\item \textbf{Page 172, proof of Corollary 3.3.26:}
``Propostion'' should be ``Proposition''.

\item \textbf{Page 174:}
``several vectors spaces'' should be ``several vector spaces''.

\item \textbf{Page 178:}
``linear maps from $V$ and $W$'' should be
``linear maps from $V$ to $W$'', and
``preceeding'' should be ``preceding''.

\item \textbf{Pages 179--180, dual-basis discussion and Proposition 3.4.2 (b):}
Replace both occurrences of ``is a a basis'' by ``is a basis''.

\item \textbf{Page 183:}
``subpace'' should be ``subspace'', and the stray period after
``Corollary 3.4.9'' should be deleted.

\item \textbf{Page 187:}
``Similarly is an equivalence relation'' should be
``Similarity is an equivalence relation''.

\item \textbf{Page 190:}
``ususal'' should be ``usual''.

\item \textbf{Page 194:}
``post--mulitplication'' should be ``post--multiplication''.
 % and
% ``$d_i0s$'' should be ``the $d_i$'s''.

\item \textbf{Page 195:} ``it's own inverse'' should be
``its own inverse''.

\item \textbf{Page 195, Smith-normal-form algorithm:}
``If there is a element $\beta$'' should be ``If there is an element
$\beta$''.

\item \textbf{Page 198, proof of Theorem 3.5.13:}
``a basis of $Z^n$'' should be ``a basis of $\ZZ^n$''.

\item \textbf{Pages 202--204:}
Replace ``is a also an $\ZZ_p$-vector space'' by ``is also a
$\ZZ_p$-vector space''; replace ``an $\ZZ_p$-vector space'' by ``a
$\ZZ_p$-vector space''; replace ``irreducible'' by ``prime'';
and replace ``occuring in an prime factorization''
by ``occurring in a prime factorization''.

\item \textbf{Page 209:}
``We have already have observed'' should be
``We have already observed'', and
``decompostion'' should be ``decomposition''.

\item \textbf{Page 212, Remark 3.6.26:}
``the group of units of $K^*$'' should be
``the multiplicative group $K^*$'' or ``the group of units of $K$''.

\item \textbf{Page 213:}
Both references to ``Lemma 3.6.25'' should be to
``Theorem 3.6.25''.

\item \textbf{Page 216, first paragraph:}
The paper models are in Appendix F, not Appendix E.  Also replace
``patterns for making paper model'' by ``patterns for making paper
models''.

\item \textbf{Page 239, first paragraph of Section 4.5:}
Delete one occurrence of ``by'' in
``implemented by by a linear isometry''.

\item \textbf{Page 243, Definition 5.1.5:}
The two proposed formulations of transitivity are equivalent only when
$X$ is nonempty.  If $X=\varnothing$, there are no orbits, whereas
``for any two elements $x,x'\in X$'' is vacuously true.  Add the
hypothesis $X\neq\varnothing$, or choose one of the two formulations
as the definition.

\item \textbf{Page 245, Definition 5.1.16:}
Replace ``$\operatorname{Cent}_G(x)$''
by ``$\operatorname{Cent}_G(g)$''.

\item \textbf{Page 253, Remark 5.3.2:}
``The rather unexpected answer that it is true'' should be
``The rather unexpected answer is that it is true''.

\item \textbf{Page 254, Exercise 5.3.5:}
``determined by rational $2$-by-$2$ matrix'' should be
``determined by a rational $2$-by-$2$ matrix''.

\item \textbf{Page 258, Theorem 5.4.9:}
``there is a $a\in G$'' should be ``there is an $a\in G$''.

\item \textbf{Page 265, proof of Lemma 5.5.1:}
``the intersections of the stabilizers'' should be
``the intersection of the stabilizers''.

\item \textbf{Page 265, Remark 5.5.2:}
``$A_n$ is the only nontrivial normal subgroup of $S_n$''
should read ``$A_n$ is the only nontrivial \emph{proper} normal
subgroup of $S_n$'', since $S_n$ itself is also a nontrivial normal
subgroup.

\item \textbf{Page 284, Proposition 6.2.27 (b):}
``interesection'' should be ``intersection''.

\item \textbf{Page 289, quotient-ring discussion:}
Delete one occurrence of ``this'' in ``check that this this is well
defined'', and insert a space in ``degree of $f$,the product''.

\item \textbf{Page 305, Lemma 6.5.18:}
``properties of an nonzero nonunit element'' should be ``properties of a
nonzero nonunit element''.

\item \textbf{Page 310, Corollary 6.6.5 and Example 6.6.6:}
Replace ``unique factorization domain domain'' by ``unique factorization
domain'', ``an nonzero'' by ``a nonzero'', and ``In an UFD'' by ``In a
UFD''.

\item \textbf{Page 355, Definition 8.1.26:}
The subscript ``$s$'' on the left hand side of the
first equality should be ``$n$''.

\item \textbf{Page 363, Lemma 8.3.3:}
``acts $S_n$'' should be ``$S_n$ acts''.

\item \textbf{Page 363, proof of Lemma 8.3.3:}
In the long computation, replace
\newline
``$x_{\sigma(\tau(1))}, \ldots, x_{\sigma(\tau(1))}$'' by
``$x_{\sigma(\tau(1))}, \ldots, x_{\sigma(\tau(n))}$''.

\item \textbf{Page 375, proof of Lemma 8.4.1:}
In ``Each $w_j$ has a unique expression as an $R$–linear
combination of the basis elements $v_j$'', it would be
better to replace``$v_j$'' by ``$v_i$''.
Replace ``as an $R$–linear combinations''
by ``as an $R$–linear combination''.
In the formula for $v_j$ (between (8.4.1) and (8.4.2)),
replace ``$b_{n,j}w_n$'' by ``$b_{m,j}w_m$''.
Finally, in the last sentence of the proof,
``two basis'' should be ``two bases''.

\item \textbf{Page 379, definition of length:}
The period before ``where $u$ is a unit and the $p_i$'s are irreducibles''
should be a comma.

\item \textbf{Page 382, proof of Lemma 8.4.11:}
``as an $R$–linear combinations'' should be
``as an $R$–linear combination''.

\item \textbf{Page 399, paragraph after the block-diagonal display:}
Delete one occurrence of ``subspace'' in ``the invariant subspace
subspace $V_i$''.

\item \textbf{Page 410, Definition 8.6.14:}
``an \textit{nonzero} vector'' should be ``a \textit{nonzero} vector''.

\item \textbf{Page 417, Definition 8.7.7:}
``similar $A$'' should be ``similar to $A$''.

\item \textbf{Page 435, proof of Proposition 9.4.4:}
There is a closing parenthesis
too much in ``$k_n\sigma(\alpha^n))$''.

\item \textbf{Page 453, Exercise 9.6.15:}
The ``g'' should be a mathmode ``$g$''.

\item \textbf{Page 457:}
Replace ``$(b_j/b_n)$'' by ``$(b_j/b_m)$''
(shortly after (9.7.2)).
In the next sentence, replace ``and the total
degree as a polynomial in the $(b_j/b_m)$ is $m$''
by ``and the total
degree as a polynomial in the $(b_j/b_m)$ is $n$''.

\item \textbf{Page 466, proof of Lemma 9.8.5:}
``By by the Galois correspondence'' should be ``By the Galois
correspondence''.

\item \textbf{Page 498, Exercise 11.1.1:}
``there is an most one point'' should be ``there is at most one point''.

\item \textbf{Page 547, Definitions E.1 and E.2:}
``A linear combination of set $S$'' should be ``A linear combination of a
set $S$'', and ``A set $S$ vectors'' should be ``A set $S$ of vectors''.

\item \textbf{Page 548, first sentence:}
``a linear independent set'' should be ``a linearly independent set''.

\item \textbf{Page 548, second sentence:}
``The empty set is linearly independent, since there are no
sequences of its elements''. There are! But only the empty one, so
the conclusion still holds.

\item \textbf{Page 550, first paragraph after Definition E.6:}
The map is printed as $x\mapsto =Mx$; delete the extraneous equals sign.

\item \textbf{Page 551, first line:}
The identity matrix $E_n$ acts on $K^n$, not on the undefined $K^N$.

\item \textbf{Page 551, second paragraph:}
``$\operatorname{Hom}_k(K^n,K^m)$'' should be
``$\operatorname{Hom}_K(K^n,K^m)$''.

\item \textbf{Page 552, second paragraph:}
Replace ``$T(\boldsymbol{x}) = \sum \alpha_i T(\boldsymbol{e}_i)$''
by ``$T(\boldsymbol{x}) = \sum \alpha_i T(\widehat{\boldsymbol{e}}_i)$''.

\item \textbf{Page 555, second paragraph:}
``A matrix has a left inverse'' should be
``A matrix $M$ has a left inverse''.

\item \textbf{Page 556, definition of an inner product:}
The third property ``$\left< \boldsymbol{x},
\boldsymbol{x}\right> \geq 0$ and $\left< \boldsymbol{x},
\boldsymbol{x}\right> = 0$ if, and only if
$\boldsymbol{x} = \boldsymbol{0}$'' is confusingly
worded. Of course, what is meant is that
$\left< \boldsymbol{x}, \boldsymbol{x}\right> \geq 0$
holds for each vector $\boldsymbol{x}$, whereas
$\left< \boldsymbol{x},
\boldsymbol{x}\right> = 0$ holds if and only if
$\boldsymbol{x} = \boldsymbol{0}$.

\item \textbf{Page 556, Section E.3:}
``Cauchy--Schwartz'' should be ``Cauchy--Schwarz'' (twice).

\item \textbf{Page 557, first paragraph:}
``$\boldsymbol{v}_i,\ldots,\boldsymbol{v}_s$'' should be
``$\boldsymbol{v}_1,\ldots,\boldsymbol{v}_s$''.

\item \textbf{Page 557, third paragraph:}
``$\{\boldsymbol{v}_i,\ldots,\boldsymbol{v}_n\}$'' should be
``$\{\boldsymbol{v}_1,\ldots,\boldsymbol{v}_n\}$''.

\item \textbf{Page 557, fourth paragraph:}
Delete one occurrence of ``to'' in ``$\boldsymbol{w}_j$ is orthogonal to to
$B_j$''.

\end{enumerate}

\end{document}
