Remarks and Errata to the Second Edition of "A Concrete Introduction to Higher Algebra",
by Lindsay N. Childs, published by
Errata (as of
These are corrections to the softcover edition, published in 2000. Names in parentheses following the errata refer to the discoverer. My thanks to Donald Crowe of the University of Wisconsin, Madison, Richard Ehrenborg, Margaret Readdy and Chris Jeuell of Cornell, and Olav Hjortås of Bergen for sending me many of the errata below.
Any additional errata will be greatly appreciated. Please send them to LC802@math.albany.edu
p. 8, line -6, "Then P(n) is true..."
p. 9 (line -14): Delete "(1)". (Chris Jeuell)
p. 9 (line -5): "n uses" should be replaced by "n - n_0" uses". (Jamie Bessich)
p. 10 (line -12): The expression (k^3 + 3k^2 + 3k + 1) should not itself be cubed. Similarly, (k^2 + 2k + 1) should not be squared.(Chris Jeuell)
p. 12, E12 is defective and should be omitted (it appears impossible to prove either part by induction).
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. 16 (line -7):
Replace "Example 6, above" by "Example 1' (from 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. 26, line -17. The common divisors of 15 and 42 are 1 and 3.
pp. 31 - 32. Margaret Readdy writes: "Use matrix notation to give a nicer presentation for gcd and Bezout."
p. 32 (line -10): q should be q_1, as in: r_1 = b×1 + a×(-q_1).(Chris Jeuell)
p. 36 (line -6): "...after 128 divisions ..." (Chris Jeuell)
p. 37 (line 12): "If the quotients q_1, q_2, ... q_n are large..."(Chris
Jeuell)
P. 38 line 10 "for any n > 1, a_{n+1} = a_n + a_{n-1}" (Margaret Readdy)
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. 54, E24: "Using the last two exercises and E20, prove..."
p. 55 (E1): In the hint, p1 should be italicized.(Chris Jeuell)
p, 58, line -15, "by Proposition 2" (not Proposition 1),(found by Morris Orzech)
p. 67 E9 (ii) "(a_1 + ... + a_n)^2 \cong..."(Margaret Readdy)
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 solution with |x| minimal is...” (Bill Hammond)
p. 80, E2: change the reference to (iiia), p. 66.
p. 80, -15, in the second line of the proof of (ii) of Proposition 1, change = to congruence.
p. 82, line 3 should read: if a \equiv a’ (mod m) and b \equiv b' (mod m), then a + b \equiv a' + b' (mod m).
page 88, line 2. “... we call a a unit...”--the first “a” should be in italics.
page 97, line -3 should note that M is prime, as in “If M is prime, then a number A whose order modulo M is exactly M-1 is called a primitive element...” (Bill Hammond)
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 in cases where 0, a, b and a+b are not all distinct. In Z/4Z, 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, line 3, "Find the order of i+1".
p. 127, line -5, the (iii) should be on the left margin.
p. 129, line 2 of proof of Proposition 2: "If f(a) = 0, then 1 =" (Olav Hjortaas has very sharp eyes!)
p. 135, lines 6, 7, 13 and -6, roman "a" should be italic "a", also on line 1 of the next page. (The typesetter changed virtually all the italic "a"s within the text in the final manuscript to roman, and I obviously didn't find them all!)
p. 141 (E8): The expression p - 1! should be (p - 1)!(Chris Jeuell)
p. 142, line 2. “... be the number of units of Z/mZ, where m >= 2. Then...”
p. 143, line above E3: "set of units of Z/mZ"
p. 144, lines 1-5: We define phi(m) here to be the number of numbers r with 1 <= r <= m that are relatively prime to m. In particular, phi(1) = 1. This definition then extends to m = 1 the definition of phi(m) on page 142, line 2, and avoids confusion about what phi(1) should be. (As David Ford pointed out, we need phi(1) = 1 for E11 on that page.)
p. 153, line -12, m < g_0 should be m \le g_0.
p. 157, line -4: = (.46,58,17,8,34,17,8,...)
p. 161, line -11: r/s = .q_1 ...
Section 10B, p. 164ff. Margaret Readdy writes: "There is no need to put all of these restrictions on RSA. There are four cases, namely (w,N) = 1, (w,N) = p, (w,N) = q and (w,N) = N. The second and third cases are not hard to prove for
the students and thus one has the full strength of RSA available. See Rivest, Shamir and Adleman article in Communications of ACM." See also Section 26F.
p. 169 lines -9, -8 "...then for any integer a [italic] relatively prime to n, n divides a^{n-1} - 1.(Margaret Readdy)
p. 169. 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 observed that once encoded, the message cannot be decoded!
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. 172, Proposition 2: "... then m = ... is a perfect number" (all in italics)
p. 196 (line
12): "Then x = 3x_1 + 6x_2 + (-1)x_3 =
-3946..." (Chris Jeuell)
p. 200, line -4: "0 \le a_2 \le
m_2"
p. 223, just above E1 : ‘‘The receiver will be
misled only if there are 3 or more errors.’’
(Ken Brown)
p. 240, line 13, s > 0 should be s \ge 0.
p. 243, E12: “and a polynomial f(x)…” –the “a” should not be in italics.
p. 243, line -3, "an element", not "a element".
p. 248, line -5. Clearer would be: “A polynomial p in F[x] is irreducible if the degree of p is \ge 1 and p does not factor into the product of two polynomials each of degree < the degree of p. Thus p is irreducible if p is not a unit, and for every factorization p = fg in F[x], then f or g is a unit, that is, a constant polynomial.”
p. 249, E 10, "with f(x)r(x) + g(x)s(x) = d(x)"; E 12, "for all r \ne 0, 1" (Bill Hammond).
p. 253, line -8, the last factor of x^3 - 2 in C[x] should be x - (\omega^2)2^(1/3).
p, 264, line 2, the square root sign should not cover the x.
p. 271, line -2: D = {z : |z| \le R).
p. 279, line -5, italicize the x in "x^3".
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 x^4 - 3x^2 + 9 is factorable over the integers; to see this, write it as (x^4 + 6x^2 + 9) - 9x^2, which is the difference of two squares. (Chris Jeuell)
p. 289 line -4 Change second term to a_{n-1} r^{n-1} s.(Margaret Readdy)
p. 292, line 7: "Hence \phi_p(g(0)) = \phi_p(h(0)) = 0 [wrong subscript on the second \phi]
p. 298, line 4: "new", not "nwe".
p. 302, line -6: “congruences modulo a polynomial, …”: the “a” should not be in italics.
p. 303, line 4 of Example 2: deg r(x) < deg m(x).
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. 308, line 1 of Section B: “modulo a polynomial”: the “a” should not be in italics.
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)."
p. 317, line -10, the coefficient of x in f(x) should be (a_1 + a_3x^2)
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. 326, line 5 of Example 1(continued): r_1(x) = x^2
p. 326, line -3, "The equation (3) becomes
p. 350, line 8: the first strategy on line 9 was not used in the example above (but is useful in other examples).
p. 351, line 2: the italic a should be roman (see comment on p. 135)
p. 353, line 3 of section A: omit the ).
p. 354, Proposition 1. "A finite group G of order n is cyclic if and only if..."
p. 355, lines 2-3. Omit: "Assume in this next theorem that the group operation is multiplication"; lines 11-12: "Then a^s = a^{dq}*a^r, so a^r = a^s*(a^d)^{-q} is a product... "
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. 418, E13, line 2: "many", not "may".
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" (Margeret Readdy).
p. 431, E5, (ii): "Why does this not contradict the theorem that a polynomial of degree 4 cannot have more than four roots in a field."
p. 442, line 5: Change first alpha to a.
p. 442, line -6: Change subscript on C_l(x) to capital I (twice).
p. 442, line -5: Change minus to plus. (Margaret Readdy)
p. 444 line -3: Delete 1 in denominator of alpha + 1, that is, change
expression in parenthesis to (\alpha + 1 + 1).(Margaret Readdy)
p. 447, line -5: "... evaluates R(x) at \alpha^i, i = 1, ..., 6. Since C(\alpha^i) = 0, i = 1, ..., 6,..."
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. Prof. Readdy substituted (10110) .
p. 487, Section 6D, E1: (a) and (d) are, (b, (c) and (e) are not.
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, E7 (b)(ii), the answer should be 2\a^2 + \a + 2.
Last update, January 23, 2004