Errata in the revised 3rd edition of Lang's ALGEBRA, discovered by
Khalilah Beal, Sumit Gulwani, Joel Los, Kurt Luoto, Michael Manapat,
Bjorn Poonen, Stephanie Reynolds, Brian Rothbach, and Yuan Yao (beyond
those mentioned in Bergman's notes)
p. viii, last paragraph: Ina Lindemann's last name is misspelled.
p. xiii: The table of contents entry for Section 1 of Chapter IX
("Hilbert's Nullstellensatz") should refer to page 378, not 377.
p. 18, statement of Proposition 3.1: delete the comma before "whose"
p. 18: in the proof of proposition 3.1, it says "it suffices to
prove that if G is finite and abelian, then it admits a cyclic
tower"...it should end "admits a cyclic tower ending in
{e}", and similarly in the sentence starting with "By induction"
p. 18, example at bottom: Theorem 6.4 should be Theorem 6.5
p. 23, 3rd paragraph of section 4: "a subgroup of G, which is cyclic"
implies that G is cyclic; it should be reworded as "a cyclic subgroup of G"
p. 25, proof of (vi): change "we are reduced to the case when G is a p-group"
to "we may reduce to the case where G is a noncyclic p-group".
(The current wording suggests that if G is not cyclic,
then it is a p-group.)
p. 45, line -3: "Given y in A" -- the A should be S
p. 52, line 5: "for all n" --> "for all r"
p. 52, last line of proof of Theorem 10.1: "direct limit" --> "inverse limit"
p. 52, line -6: "This occurs when the group G is countably generated."
Not so: a countably generated group can have uncountably many subgroups
of finite index. If G is an F_2-vector space of countable dimension,
then G has uncountably many subgroups of index 2.
p. 81, Exercise 51(b):
This is ambiguous. There are at least 3 ways of interpreting this:
1) F(X x Y) = F(X) x F(Y)
Z F(Z)
where in F(Z), we consider Z as an object of C_Z via id_Z: Z --> Z.
In this interpretation, the "fiber product" on the right
is really just the product of F(X) and F(Y) in SET,
since F(Z) is a 1-element set.
2) F(X x Y) = F(X) x F(Y)
W F(W)
where X,Y,W are objects in C_Z, and the fiber product on the left is taken
in the category C_Z.
3) Maybe (although this disagrees with what is written) what was intended
was to define
F: C --> SET
X |--> Mor(T,X)
and then prove
F(X x Y) = F(X) x F(Y).
Z F(Z)
p. 83, line 2: the first "respectively" should be deleted
p. 84, p. 86: "commutative" is defined twice
p. 85, second to last display: the double sum with terms
x_i x_j should have terms x_i y_j.
p. 90, line 11: "containing 1" can be deleted
p. 90, definition of A[S]: it might be good to clarify that n is not fixed,
but ranges over nonnegative integers.
p. 90, last sentence before Example:
"elements of S may not commute with each other" is ambiguous.
Better to say "elements of S do not necessarily commute with each other".
p. 100, line 8: The sum should start with i=0.
p. 100, line -3: "polynomial" --> "polynomials"
p. 113, before first Example: "relaively prime" (typo)
p. 115, Exercise 5: The second part is not literally true as stated
(e.g. 6 is a prime in Z[1/3] but not a prime in Z).
Better would be to change the second part to
"If P is a set of representatives for the equivalence classes
of primes of A (as on p.113), then the natural map A --> S^{-1} A
maps { p in P : (p) intersect S is empty } bijectively to a set
of representatives for the equivalence classes of primes of S^{-1} A."
p. 115, Exercise 10: "set of all element" should be "set of all elements"
p. 126, line 6 of "Examples: modules over a group ring":
F(sigma) should go from F(K) to F(K').
p. 161, fifth line of "Example": change "for" to "to"
p. 162, 4th line of example, and below the last display: The term
"projective limit" is used, but only "inverse limit" has been defined.
p. 163, 3rd line of "Examples": k[T] should be k[[T]]
p. 165, Exercise 2: change "commutative ring" to "nonzero commutative ring"
p. 166, Exercise 6: This is false. For instance, G need not map M into itself.
p. 169, Exercise 17(a): The term "projective system" has not been defined.
p. 170, Exercise 17(b): This is false. The limit should be over
nonzero ideals only.
p. 172, before Exercise 27: The definition of filtered algebra given
does not imply that A is an A_0-algebra in either sense given on p.121,
because A_0 need not be commutative. Even if A_0 is commutative,
it need not be in the center of A. (Consider k=F_p, L a nontrivial
field extension of k, and A a twisted polynomial ring (twisted by the
Frobenius endomorphism of L) graded in the obvious way.)
p. 176, first display: f_i should be f_j
p. 176, line before third display: f_i should be f_j
p. 177: 2nd line of proof of Theorem 1.9: It would be better
to cite Proposition 4.3(v) instead of Proposition 4.3(vi).
p. 179, Proposition 1.12: The sum should start with nu=0.
And in the next display, after the proof, the sum should start with mu=0.
p. 186, first line of proof of Theorem 4.1:
The brackets in "A[X]" should not be italicized.
p. 187, last sentence of the proof of Theorem 4.1:
This repeats what was said in the preceding sentence,
instead of completing the proof. It should be changed to
"By induction, the polynomial f - c_1 f_{d1} - ... -
c_{n_d} f_{d n_d} lies in the ideal generated by the
f_{ij}, from which we conclude that f does as well."
p. 193, line 5 of Cubic case: Theorem 6.1 as stated implies only that
the weight is <=6.
p. 193, line after display defining f_1(x):
Should be D = va^3 = -v, not D = -va^3 = -v.
p. 193, line after display defining f_2(x):
Should be D = wb^2 = w, not D = 2b^2 = w.
p. 213, Exercise 7(b): p should be defined as the characteristic of k
p. 214, Exercise 8: The assumption that A is entire is needed not here,
but in Exercise 9.
p. 214, Exercise 9: One must again assume that the automorphism acts as the
identity on A.
p. 215, Exercise 12(c): change D_f to D(f) to agree with notation on p.204.
p. 216, line 2: b_{d-1} should be b_{d-2}.
p. 216, Exercise 18: Why not simplify and strengthen the statement
by deleting "sufficiently large" before part (a)?
p. 216, Exercise 18(c): This is false as stated.
Replace "sufficiently large" by "sufficiently large and positive"
in both places.
p. 229, Remark: The notation f^sigma has not been defined when
f is a polynomial.
p. 230, last line: Is p^sigma the image of p under k[X] --> k[X]/(p(X)),
or the image of p under the induced map k[X] --> k[X]/(p(X))[X] ?
It seems to be the latter, while (p(X))^sigma is the former.
It would be less confusing if one did not use the same letter X
in both roles.
p. 230, proof of Lemma 2.1: Here is a slightly simpler proof:
"Since sigma is injective, it will suffice to prove that sigma is surjective.
Let alpha be an element of E, and let f \in k[x] be such that
f(alpha)=0. Let S be the set of zeros of f. Then sigma maps S to itself.
But S is finite and sigma is injective, so sigma(S)=S.
In particular, alpha is in the image of sigma."
p. 234, proof of Corollary 2.9: replace "identity mapping on k"
with "inclusion map k --> E' "
p. 235, "a large amount of" should be "a large number of" or "many"
p. 243, statement of proof of Theorem 4.6: The special rules of grammar
pertaining to the word "number" say that
"There is only a finite number of..." should be
"There are only a finite number of...". Better still would be
"There are only finitely many"
The same comment applies in many other places, such as:
p. 263, statement and proof of Corollary 1.6
p. 322, exercise 17(b)
p. 327, exercise 37.
p. 264, statement of Theorem 1.8: add "of order n" after "let G be a
finite group" and delete "of order n" after "automorphisms of K" (to
clarify that the group G has order n--the original formulation could
have meant "Let G be a group consisting of automorphisms sigma, where
sigma^n = id")
p. 270, line 8: "Since char k not= 2,3, f is separable."
In fact, one is using only char k not= 3 (so that f' is not the zero
polynomial).
p. 274: p should be 5 in the last two sentences of Example 7.
p. 286, statement of corollary 5.4: "the distinct set of embeddings"
sounds funny. Maybe say just "the distinct embeddings".
p. 288, first and second lines after the last display
in the proof of Proposition 5.6: N_{F/k}(\alpha) is
the determinant of m_{\alpha} on F (delete the exponent)
and N_{F/k}(\alpha)^r is the determinant of m_{\alpha}
on E (change the exponent d to r).
p. 289, Theorem 6.2: The assumption "n an integer >0 prime to the
characteristic of k" rules out characteristic 0. It should be
replaced by "n an integer >0 not divisible by the characteristic of k".
p. 290, first sentence of the proof of Theorem 6.3: Although this
proof is easy, it is confusingly written. The first "trace" is the
trace of beta, whereas the second trace is presumably the trace of
alpha. Maybe it would be clearer to say explicitly that
Tr(alpha)=Tr(sigma alpha), and then deduce Tr(beta)=0.
p. 293, third line before the Remark: add "of" before "prime degree"
p. 289, Theorem 8.1: The assumption "m an integer >0 prime to the
characteristic of k" rules out characteristic 0. It should be
replaced by "m an integer >0 not divisible by the characteristic of k".
p. 303, second line of the proof of Lemma 10.2: Add a ")" to the end of the line.
p. 303, second line from bottom: "so f is a coboundary" should be
"so (\tau - 1)f is a coboundary"
p. 322, problem 8(b): Add "of" before "D_8 \subset S_4"
p. 323, "Note" starting after Problem 20: There is no closing bracket
("]") ending the note. Probably it should be placed after the
reference [Sch 58] on the following page.
p. 325, exercise 23(b): The statement is false when G is the trivial group.
p. 326, Problem 34: The "respectively" doesn't make any sense here
and should be deleted. In its place, perhaps one could add "each of"
before "two distinct subfields". Also, there is no need for the word
"distinct".
p. 603, last line: "over k" should be "over R"
p. 611, just before the end of 1st paragraph of proof:
Change the comma after "id" to a period.
p. 611, line -2: Change x'' to x and E'' to E.
p. 877, line 18 from top: change x(subscript v) to x(subscript y).
p. 880, line 6: "let us take Example 2" should be "let us take Example 1"
p. 880, 8 lines from the bottom: In "restriction of u to J",
the J should be J'.
p. 885, line 9: "card(A" should be "card(A)"
p. 885, the passage beginning with line 7 from the top and ending with
"Given two nonempty sets A, B..." (line 15) is oddly repetitive.
p. 886, line 4: "H(f^{-1}(b))" should be "h(f^{-1}(b))
p. 888, line 15: "f: B x B is a bijection"--delete the ": B x B"
p. 888, line 16: "on" should be "onto"
p. 892, line 10 (proof of theorem 4.1): Lang never defines "initial segment"
before using it here (at least not in the appendix)
p. 892, line 10: "the initial segment" should be "an initial segment"
p. 893: the second sentence of problem 9 -- "containing K as a subfield
and algebraic over K" would perhaps be better as
"that contains K as a subfield and is algebraic over K"