Abstract Algebra with a Concrete Introduction
(ISBN 0-13-004425-3) was published in 1990 by Prentice-Hall, Inc. It is now out of print, but some copies are available on the web. This file contains the table of contents and an errata sheet.

#### Chapter 1 Integers

1.1 Divisors
1.2 Primes
1.3 Congruences
1.4 Integers Modulo n
Appendix A: Sets
Appendix B: Construction of the Number Systems
Appendix C: Basic Properties of the Integers
Appendix D: Induction

#### Chapter 2 Functions

2.1 Functions
2.2 Equivalence relations
2.3 Permutations
Appendix E: Complex Numbers

#### Chapter 3 Groups

3.1 Definition of a Group
3.2 Subgroups
3.3 Constructing Examples
3.4 Isomorphisms
3.5 Cyclic Groups
3.6 Permutation Groups
3.7 Homomorphisms
3.8 Cosets, Normal Subgroups, and Factor Groups

#### Chapter 4 Polynomials

4.1 Fields; Roots of Polynomials
4.2 Factors
4.3 Polynomials with Integer Coefficients
4.4 Existence of Roots
Appendix F: Solution by Radicals of Real Cubic and Quartic Equations

#### Chapter 5 Commutative Rings

5.1 Commutative Rings; Integral Domains
5.2 Ring Homomorphisms
5.3 Ideals and Factor Rings
5.4 Quotient Fields

#### Chapter 6 Fields

6.1 Algebraic Elements
6.2 Finite and Algebraic Extensions
6.3 Geometric Constructions
6.4 Splitting Fields
6.5 Finite Fields
Appendix G: Dimension of a Vector Space

#### Chapter 7 Structure of Groups

7.1 Isomorphism Theorems; Automorphisms
7.2 Conjugacy
7.3 Groups Acting on Sets
7.4 The Sylow Theorems
7.5 Finite Abelian Groups
7.6 Solvable Groups
7.7 Simple Groups

#### Chapter 8 Galois Theory

8.1 The Galois Group of a Polynomial
8.2 Multiplicity of Roots
8.3 The Fundamental Theorem of Galois Theory

#### Bibliography

approximately 350 pages

#### Errata


NOTE:  The corrections are written using LaTeX commands.
The latest corrections were added in May, 1997.

3, line 12
For
integers that can be represented
primes that can be represented

20, line 4
For
Example 1.2.1
Example 1.2.2

20, line -19
For
every component
every exponent

24, line 10
For
the proof of the theorem
the proof of the proposition

36, line 13
For
compute  $\varphi (24)$
compute  $\varphi (27)$

42, line 9
For
$| a_n - a_m | < 0$
$| a_n - a_m | < \epsilon$

53, line -14
For
subset of  $S \times T$
subset of  $S \times U$

56, line 9
For
$B = \{ f(x_1 ), f(x_1 ), \ldots, f(x_n ) \}$
$B = \{ f(x_1 ), f(x_2 ), \ldots, f(x_n ) \}$

65, line -10
For
correspond to the equivalence classes $S/f$
are the equivalence classes in $S/f$

67, line 8
For
Example 2.2.6
Proposition 2.2.5

72, line 8
For
It $\sigma \tau = \tau \sigma$
If $\sigma \tau = \tau \sigma$

80, line 8
For
$\frac{bd - ad}{c^2 + d^2} i$
$\frac{bc - ad}{c^2 + d^2} i$

82, line 8
Correction:
$\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2} } \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2} }$

82, line -2
Correction:
$\cos (x) = 1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$
$\sin (x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots$

83, line -16
For
never zero, it follows that $f$ is an entire function and hence
nonzero, it follows that $f$ is a bounded entire function and

84, line 1
For
$z^3 +a_2 z +a_1 z +a_0$
$z^3 +a_2 z^2 +a_1 z +a_0$

101, line -12
For
denoted by $SL_n (F)$
denoted by $SL_n (R)$

111, lines 12--13
For
$Z_n$ of integers modulo $n$
$Z_6$ of integers modulo $6$

116, line -1
For
$H = \{ , \}$
$H = \{ , \}$

128, line 17
For
$p_1 < p_2 < \ldots < p_n$
$p_1 < p_2 < \ldots < p_m$

128, line -5
For
$p_1 < p_2 < \ldots < p_n$
$p_1 < p_2 < \ldots < p_m$

132, line 22
For
Section 2.2
Section 2.3

137, line 1
For
Section 2.2
Section 2.3

142, line 18
For
$\{ x \in G \mid \phi (x) = e \}$
$\{ x \in G_1 \mid \phi (x) = e \}$

154, line -11
For
normal subgroup of $G$
subgroup of $G$

156, line 2
For
elements $a$ of order $4$ and $b$ or
elements $a$ of order $4$ and $b$ of

166, line 17
For
$\ldots + (a_2 b_0 +a_1 b_1 +a_2 b_0) x^2 +$
$\ldots + (a_2 b_0 +a_1 b_1 +a_0 b_2) x^2 +$

166, line 18
For
$c_k$  of  $f(x) g(x)$
$c_k$  of $x^k$ in $f(x) g(x)$

173, line 2
For
$q(x), r(x) \in F$
$q(x), r(x) \in F[x]$

173, line 22
For
$a_m b_{n-1} x^{m-n} g(x)$
$a_m b_n^{-1} x^{m-n} g(x)$

174, 12
For
Example 4.2.4
Example 4.2.2

175, line 3
For
give us
gives us

179, line 8
For
$x^2 + 2$
$x^2 - 2$

180, line 21
For
polynomial with rational coefficients
polynomial with rational coefficients,
such that $a_n$ and $a_0$ are nonzero

180, Exercise 11
(where $p$ is any prime number).

181, line 11
For
$\frac{c(x+d)}{(x^2+ax+b)^m}$
$\frac{cx+d}{(x^2+ax+b)^m}$

181, line -8
For
$c(x) + d(x) \mod p(x))$
$c(x) + d(x) (\mod p(x))$

181, line -7
For
$c(x)d(x) ( \mod (x))$
$c(x)d(x) ( \mod p(x))$

182, line -12
For
If $c$ is a root of $f(x)$
If $c$ is an integral root of $f(x)$

182, line -5
For
$f(n) = q(n)(c-n)$,
$f(n) = q(n)(n-c)$,

182, line -3
For
rational roots of equations such as
integer (and thus rational) roots of monic equations such as

186, line 6
For
$x^2 - 7 x^2 + 4x - 28 = 0$
$x^3 - 7 x^2 + 4x - 28 = 0$

209, line -4 and the remainder of the proof:
For
$D^{\times}$
$D^{\ast}$

211, line -8
For
Define  $+$  on  $S$
Define  $+$  on  $R_1$

225, line 5
For
domain
integral domain

239, line 18
For
polynomials in $K[x]$
nonzero polynomials in $K[x]$

239, line 19
For
polynomial $f(x) \in K[x]$
nonzero polynomial $f(x) \in K[x]$

239, line -7
For
An ideal is prime
A nonzero ideal is prime

247, line 19
For
polynomial $f(x) = a_0 +a_1 x + \ldots a_n x^n$
nonzero polynomial $f(x) = a_0 +a_1 x + \ldots + a_n x^n$

251, line -11
For
Proposition 4.2.1
Proposition 4.3.1

255, line -11
For
mapped by $\phi$ to $t(x)$
mapped by $\phi '$ to $t(x)$

257, line -4
For
finite field
finite fields

273, line 5
For
$p|Z(G)|$
$p | |Z(G)|$

288, line 1
For
adding $K_i$
adding $H$

290, line 1
For
Let  $G$  be a finite group
Let  $G$  be a group

298, line 13
For
page 298 line 13
where $\omega = (-1 + \sqrt 2 i )/2$
where $\omega = (-1 + \sqrt 3 i )/2$

303, line -4
For
simple group  $F^{\times}$
cyclic group  $F^{\times}$

309, line 12
For
$\theta \in \Gal (F/E)$
$\theta \in \Gal (F/K)$

309, line -12
For
$\omega \sqrt{2}, \omega \sqrt{2}\}$ is a basis
$\omega \sqrt{2}, \omega \sqrt{4}\}$ is a basis

314, line 2
For
$i = 2,3,\ldots,n-1$
$i = 1,2,\ldots,n-1$

314, line 8
For
If $|\Gal (F/K)| = p$
If $[F:K] = |\Gal (F/K)| = p$

314, lines 10--11
For
since $G$ is simple
since $[F:K]$ is prime

315, line 9
For
$2, \ldots, n$
$2, \ldots, m$

316, lines 2--3
For
Any element $\phi$ of $\Gal (E(\zeta) / K(\zeta))$
leaves $\zeta$ fixed, so $\phi (E) = E$
For any $\phi \in \Gal (E(\zeta) / K(\zeta))$,
$\phi (E) = E$ since $E$ is normal over $K$

316, line 6
For
$N_1 \supset \ldots \supset \{ 1 \}$
$N_1 \supset \ldots \supset \{ e \}$
prime degree $p$