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.

Table of Contents

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
8.4 Solvability by Radicals

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 
read
primes that can be represented 

20, line 4
For
Example 1.2.1
read
Example 1.2.2

20, line -19
For
every component 
read
every exponent 

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

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

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

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

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


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

67, line 8
For
Example 2.2.6
read
Proposition 2.2.5

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

80, line 8
For
$ \frac{bd - ad}{c^2 + d^2} i $
read
$ \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 
read
nonzero, it follows that $f$ is a bounded entire function and 

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

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

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

116, line -1
For
$H = \{ [1],[3] \}$ 
read
$H = \{ [1],[8] \}$ 

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

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

132, line 22
For
Section 2.2
read
Section 2.3

137, line 1
For
Section 2.2
read
Section 2.3

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

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

156, line 2
For
elements $a$ of order $4$ and $b$ or 
read
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 + $
read
$ \ldots + (a_2 b_0 +a_1 b_1 +a_0 b_2) x^2 + $

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

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

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

174, 12
For
Example 4.2.4
read
Example 4.2.2

175, line 3
For
give us
read
gives us

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

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

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

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

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

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

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

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

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

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

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

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

225, line 5
For
domain
read
integral domain

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

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

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

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

251, line -11
For
Proposition 4.2.1
read
Proposition 4.3.1

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

257, line -4
For
finite field
read
finite fields

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

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

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

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

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

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

309, line -12
For
$\omega \sqrt[3]{2}, \omega \sqrt[3]{2}\}$ is a basis 
read
$\omega \sqrt[3]{2}, \omega \sqrt[3]{4}\}$ is a basis 

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

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

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

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

316, lines 2--3
For
Any element $\phi$ of $\Gal (E(\zeta) / K(\zeta))$
leaves $\zeta$ fixed, so $\phi (E) = E$ 
read
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 \} $
read
$N_1 \supset \ldots \supset \{ e \} $

317, line -4
For
prime degree
read
prime degree $p$


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