Remarks and Errata to the Second Edition of
"A Concrete Introduction to Higher Algebra", by Lindsay N. Childs, published by
SpringerVerlag New York, 1995 (first edition, 1979), softcover printing, 2000.
Reviews
The following telegraphic review appeared in the American Mathematical Monthly,
vol. 103 (Feb. 1996), p. 282:
New edition of a durable and wellregarded text. Admirers needn't feel
concernthe essential elements and appeal of the first edition have survived
intact despite substantial revision. [A review of the first edition appeared
in the Monthly of October, 1983.] Changes include more emphasis on elementary
group theory, more treatment of computational number theory and primality
testing, new coverage of quadatric reciprocity, use of the discrete
Fourier transform.
Errata (as of 12/17/04)
These are corrections to the
softcover printing of the second edition (2000).
Names in parentheses following the errata refer to the discoverer.
My thanks to Donald Crowe of the University of Wisconsin, Madison,
Richard Ehrenborg, Chris Jewell, Ken Brown and Margaret Readdy of Cornell,
Olav
Hortås
of Bergen, TatHung Chan of Fredonia, Morris Orzech of Queen's
University and Keith
Conrad of the University of Connecticut for sending me errata in previous
printings
as well as many
of the errata below. (Keith also sent errata to the errata!)
Any additional errata will be greatly appreciated. Please send them to
childs@math.albany.edu

p. 3 (line 6): "if (a, b) is a pair with 2a = b, then 4a not = 3b, and
if
4a = 3b, then 2a not = b."

p. 9 (line 3): Label this equation by (1).

p. 9 (line 5): "n uses" should be replaced by "n  n_0" uses". (Jamie
Bessich)

p. 10 (line 12): The expression (k3 + 3k2 + 3k + 1) should not itself be
cubed. Similarly, (k2 + 2k + 1) should not be squared.(Chris Jeuell)

p. 12 (E14): "theorem" and "proof" should be in quotes. (Chris Jeuell)

p. 14 (line 10): "for some number r" should be replaced by "for some
natural number r".

p. 14 (line 14): "Suppose P(k) is true for all k, ..." (Matthew Bridiga)

p. 16 (line 7): Replace "Example 6, above" by "Example 1 of section
2B".(Chris Jeuell)

p. 20 (line 2): a/b should be replaced by b/a in the floor and fraction
expressions.(Chris Jeuell)

p. 20 (line 8): "7 + 10" should be "7 [times] 10" in the decimal expansion
of 1976.(Chris Jeuell)

p. 24 (E4): Italicize "n" in "Which n would be appropriate?".(Chris
Jeuell)

p. 26 (line 10): Italicize "a" in "to mean a does not divide ...".(Chris
Jeuell)

p. 32 (line 10): q should be q1, as in: "r1 = b X 1 + a X (q1)".
(Chris
Jeuell)

p. 36 (line 6): "...after 128 divisions ..." (found by Chris Jeuell)

p. 37 (line 12): "If the quotients q_1, q_2, ... q_n are large..."(Chris
Jeuell)

p. 39 (sentence before the Corollary): "... with d digits satisfies a <
a_{5d+2}...."(Chris Jeuell)

p. 45, line 22 and E4. This is slightly incorrect. The algorithm applied
to sqrt(19) shows that d can be larger than sqrt(19). Suppose 0 < c <
sqrt(m), 0 < d < sqrt(m) + c and d divides m  c^2, and one performs a
step of the algorithm, namely: write (sqrt(m) + c)/d = q + r where 0 < r
< 1, then rationalize the denominator of 1/r to get (sqrt(m) + c')/d'.
Then 0 < c' < sqrt(m), d' divides m  c'^2, and 0 < d' < sqrt(m) + c', not
d' < sqrt(m). So always 0 < d' < 2sqrt(m). Hence the number of pairs (c,
d) which can arise in the algorithm is bounded by 2m. Thanks to Bill
Hammond for finding this error.

p. 51 Proof of Thm 1: "If not, sqrt(2) = b/a, with a, b natural numbers.
Multiplying both sides by a and squaring, we get..." (Margaret Readdy)

p. 55 (E1): In the hint, p1 should be italicized.(Chris Jeuell)

p. 67 E9 (ii): "(a_1 + ... + a_n)^2 \cong..."(Margaret Readdy) (Actually,
the exercise is correct without the exponent 2 but the exponent 2 was
obviously intended here.)

p. 70 (E7): "Use E6 to prove the following result..."

p. 71 (line 13): Wiles' proof appeared in the Annals of Mathematics,
May,
1995. The story of how it was proved has been the subject of several
popular books. One is: Fermat's Enigma, by Simon Singh, Walker & Co.,
1997. An excerpt from Singh's book appears in the February, 1998 issue of
Math Horizons, the MAA journal about mathematics for undergraduates.

p. 73 (line 15): Change "The smallest solution" to "The smallest
solution with x minimal" (Bill Hammond)

p. 88 (line 2): "we call a a unit..."  the first "a" should be in
italics.

p. 97 (line 3): "If M is prime, then a number A whose order modulo M is
exactly M1 is called a primitive element or a primitive root modulo M."
(Bill Hammond, Keith Conrad)

p. 106 (line 10): "with 1 <= m < n < 2 sqrt(p)" (Chris Jeuell)

p. 109 (E3): "(iv) Find other values of x_0 for which N divides x_{12} 
x_6, and values of x_0 for which N doesn't divide x_{12}  x_6."(Chris
Jeuell)

p. 119 (line 10): 'c' should be italicized.(Chris Jeuell)

p. 119 (line 8): 'a' should be italicized in "for all a".(Chris Jeuell)

p. 123, Proposition 2: TatHung Chan of Fredonia observed that the ring
Z/4Z contradicts Proposition 2 on p.123. The proof breaks down because
the only values for a and b are a = b = 2, so that a, b, a+b, 0 reduce to
2, 2, 0, 0. Richard Ehrenborg and Steve Chase noted that the ring
F_2[x]/(x^2) is the only other counterexample.

p. 123 (line 10): In "if such a number b exists", the "a" should not be
italicized.(Chris Jeuell)

p. 126 (line 6): The last 'a' before '(s factors)' should be
italicized.(Chris Jeuell)

p. 127 (lines 2 and 3): In line 2, there is an implicit assumption that
the existence of inverses for nonzero elements is the
only field axiom that does not follow immediately from the fact that
polynomials with coefficients in a field (such as Z/3Z) is a
commutative ring. Kirby Baker pointed this out. In line 3, "Find the
order of
i+1".

p. 127 (E9): As in E8, i^2 = 1. ((Kirby Baker)

p. 127 (line 5): Property (iii) should be placed on the left margin.

p. 128 (line 14): "0 = f(0) = f(r + (r)) = f(r) + f(r)"

p. 129, line 2 of proof of Proposition 2: "If f(a) = 0, then 1 =" (Olav
Hortås
has very sharp eyes!)

p. 138 (E15): "2^41 is congruent to 82".
p. 141 (E8): The expression p  1! should be (p  1)!(Chris Jeuell)

p. 142 (line 2): In Euler's Theorem, require m > 1.

p. 144 (lines 15): "... The function phi is called Euler's phi function.
For m > 1, phi(m) is the number of numbers r with 1 <= r <= m that are
relatively prime to m. In particular, phi(1) = 1.
In order to use
Euler's theorem, ..." (David Ford pointed out that phi(1) = 1 is needed in
E11 on page 144.)

p. 164ff, Section 10B. Margaret Readdy writes that there is no
need to restrict the word w to be relatively prime to m: w can be any
number < m. As she points out,
there are three cases, namely (w,m) = 1,
(w,m) = p and (w,m) = q, and it's not hard for students to show that
encoding
and decoding
works in the second and third cases. (A more general remark may be
found as Proposition 1 on
page 392RSA works for all w < m if the numbers p and q are primes or
Carmichael numbers.)

p. 168 (line 8): "then using the pair (m_B, e_B)."

p. 169 (line 4:. An interesting commentary on Hardy's remarks appears in a
paper
of the eminent number theorist L. J. Mordell: Hardy's "A Mathematician"s
Apology", American Mathematical Monthly, vol. 77 (1970), pp. 831835.

p. 169, Exercise 3. Saab Yaqub Hassan of Cornell was the first to
observe that once
encoded, the message cannot be decoded!

p. 169 (lines 9, 8): "...then for any integer a [italic] relatively
prime
to n, n divides a^{n1}  1.(Margaret Readdy)

p. 171 (line 5):
For
recent information on Mersenne primes, check
http://www.mersenne.org/
and http://www.utm.edu/research/primes/notes/3021377/

p. 196 (line 12): "Then x = 3x_1 + 6x_2 + (1)x_3 = 3946..." (Chris
Jeuell)

pl 223 (line just above E1): "The receiver will be misled only if there
are 3 or more errors." (Ken Brown)

p. 235 (line 1): "coefficient".
p. 247 (line 4): "... can be written as d = rf + sg ..." (Christopher
Sytsma)

p. 251 (E5): "In F[x], F any field"both "F"s should be the same.

p. 279 line 3): "that is, a nonzero constant polynomial."

p. 287, line 2. Clearer language would be: "Suppose f(x) = a(x)b(x)
where a(x) and b(x) are in Q[x]. Then ..."

p. 288 (line 5): The coefficient of x^2 should be 4, not 3, as the
computations below show. The polynomial x4  3x2 + 9 is factorable over
the integers; to see this, write it as (x4 + 6x2 + 9)  9x2, which is the
difference of two squares. (Chris Jeuell)

p. 289 (line 4): Change second term to a_{n1} r^{n1} .(Margaret Readdy)

p. 302 (line 6): the "a" should not be in italics.

p. 303 (line 4): "sides of the congruence) x is congruent to 1 (mod x1).
Raising both sides..." (Keith Conrad)

p. 303 line 12 In Exercise 1 change "root" to "remainder".(Margaret
Readdy)

p. 305 line 1. The (*) refers to the equation at line 8, which should
be labeled with (*)(Margaret Readdy)

p. 306 (line 8): Delete extra comma.(Margaret Readdy)

p. 309 (E2 (ii)): The first modulus should be x^4 + x + 1, not x^4 + x +
x.
(Margaret Readdy)

p. 313, Section 21B. Margaret Readdy writes: "As a comment, I showed
the 4 x 4 and 8 x 8 Fast Fourier Transform matrix decompositions to my
class, as well as explained butterfly diagrams to them (and piping, to
speed up the process)." To this comment, Keith Conrad wrote, "I
don't see how this is a comment that belongs in an errata list".
I've left it in as a challenge for me to someday understand what the
comment means!

p. 320 (fifth line after the matrix): the second entry of the vector
should be omega^{i}, not omega^{1}.(Chris Jeuell)

p. 322 (E6 (ii)): the (i,j)th entry should be 5^{ij}. Also, the notation
"(i  j)th entry" should be "(i,j)th entry".(Margaret Readdy)

p. 335 (line 8): b_{r3}

p. 335 (line 13): too many commas

p. 350 (line 8): the first strategy on line 9 was not used in the
example
above (but is useful in other examples).

p. 355 (line 10): "...distinct. To do this"

p. 355 (line 7):
"d divides n...."

p. 369 (line 22): Comma after a^r.(Margaret Readdy)

p. 416 (E2): "representative"

p. 421 (E4): There should be a left parenthesis after the slash (i.e.,
F_2[x]/(x^4 + ..).(Chris Jeuell)

p. 429 (line 8): The left parenthesis in p(x) should not be italicized.
(!)(Chris Jeuell)

p. 431 (E5 (i)): Change "root" to "roots". Delete comma after "are"
Margaret Readdy).

p. 442 (line 6):
Change subscript on C_I(x) to capital I
(twice).

p. 444 (line 3): delete 1 in denominator
of alpha + 1, that is, change expression in parenthesis to (\alpha + 1 +
1).(Margaret Readdy)

p. 448, Conditions for rank S, starting at line 11: Margaret Readdy
writes: "The "shortcut" conditions for the rank of S are really wrong.
You cannot compute the rank of a matrix by looking at the upper lefthand
square minors. For example, in Code V if you have error polynomial E(x) =
x^11 + x^8 + x^7, one has E(\alpha) = 0, and thus no errors, by the
shortcut condition for rank."

p. 450 (E19 a): Missing digit. We substituted (10110).(Margaret Readdy)

p. 487 (Section 6D, E1): (a) and (d) are, (b), (c) and (e) are not.

p. 488 (line 2, E1): [7], not [6].

p. 494 (line 5): E4 (a) should be x^4 + x^2 + 1, and E5 (b) should be
2x^5
+ 2x^3 + 2x^2 + 2.

p. 505 (section 28B, E7 (b)(ii)): the answer should be 2 alpha^2 + alpha +
2.
Last
update, December 18, 2004