Remarks and Errata to the Second Edition of "A Concrete Introduction to Higher Algebra", by Lindsay N. Childs, published by Springer-Verlag New York, 1995 (first edition, 1979), softcover printing, 2000.

Reviews

New edition of a durable and well-regarded text. Admirers needn't feel concern--the 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, Tat-Hung 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 M-1 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: Tat-Hung 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 non-zero 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 1-5): "... 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 392--RSA 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. 831-835.
• 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^{n-1} - 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 non-zero 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_{n-1} r^{n-1} .(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 x-1). 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_{r-3}
• 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 left-hand 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