Misprints

There is a gap in the proof of the Fundamental Theorem of Galois Theory (Theorem 8.24, page 258), pointed out to me by Gary McGuire. Here is a corrected proof.

Let K/F be a Galois extension with Galois group G. The proof given shows that Fix(Gal(K/L))=L for all intermediate fields L. To show that the maps Fix and Gal are inverse bijections, we also need to show that Gal(K/Fix(H))=H for all subgroups H of G.

We need the forward implication in the Theorem of the Primitive Element, Theorem 8.34: if K/F is a finite extension and only finitely many fields lie between F and K, then K=F(a) for some a in K. The proof of this uses no Galois theory. Now it follows from what is proved that, if K/F is a Galois extension, then the map from intermediate fields to subgroups is injective; so we conclude that K=F(a) for some a, since the finite group G has only finitely many subgroups.

Let H be any subgroup. Since K is a Galois extension of Fix(H), with Galois group H', say, it will suffice to show that if H<H' then Fix(H) is strictly larger than Fix(H'). So we have to prove the following:

Lemma. Let K/F be a Galois extension with Galois group G. If H<G, then Fix(H)>F.

Proof. Suppose, for a contradiction, that Fix(H)=F. Let K=F(a) have degree n over F, and let a=a₁, ..., a_n be the roots of the minimal polynomial of a over F. Now G permutes {a₁, ..., a_n} transitively, and hence regularly. So the proper subgroup H cannot act transitively on this set. Let {a₁, ..., a_r} be an orbit. Then the coefficients of the monic polynomial with roots a₁, ..., a_r, being the elementary symmetric functions in these values, are all fixed by H, and so lie in F.Thus a=a₁ satisfies a polynomial of degree r over F, contradicting the fact that [F(a):F]=n. So we are done.

Other misprints and corrections:

• On Page 5, in the example, the intersection of A and B should be {2} and their symmetric difference {1, 3, 4, 5}.
• Page 45, line 8, has an extra ) that shouldn't be there.
• Page 45, line 18 should say "only finitely many are non-zero" (rather than "only finitely many are zero"). (Spotted by Young-Han Kim.)
• Page 50, second line of proof of Theorem 2.15: a(1-cd)=1 should read a(1-cd)=0. (Spotted by Amjad Tuffaha.)
• "Maximal ideals": Bill Martin points out two problems: First, they are used (in Exercise 2.15, page 46) before they are defined; second, the definition doesn't make clear that a ring is not a maximal ideal of itself!
• Page 63, line 9: ab in X. (Spotted by Roderick Forman.)
• Page 72, line 18: h should be H. (Spotted by Nayim Rashid.)
• Page 73, line 2, a,b subsets of G should read A,B subsets of G. (Spotted by Laura Alexander).
• Page 74, line beginning (E1): g¹ should be g₁. (Spotted by Elizabeth Rothwell.)
• Page 79, line -13. The thetas should be on the right. Similarly on Page 83, lines 5-6. (Spotted by Roderick Forman.)
• Page 80, proof of Proposition 3.11(a): the conclusion should be that Im(theta) is a subgroup of H (not G). (Spotted by Ben Rubin.)
• Page 80, line -7: R should read G.
• Page 82, line 15: the left hand side of the equation should be ((Ng₁)(Ng₂))(Ng₃), not ((Ng₁)(Ng₂))(Ng₂). (Spotted by Jessica Hubbs.)
• Page 83, line 3: it should say "for all g in G". (Spotted by M. Q. Baig.)
• Page 84, displayed equation defining centraliser of x: the quantifier "for all g in G" is incorrect and should be deleted. (Spotted by Laura Alexander.)
• Ioannis Pantelidakis suggests a simpler argument for Step 2 of Theorem 3.33 on page 102:

  Step 2: Given a generating set S for G, if we know the product of each element of G by each element of S then G has been determined. This is because we can obtain the rest of the multiplication table by the associative law. So we can only count the number of tables of size n × log₂n and there are at most n^(n log₂n) of them.

• Page 104, vector space axioms: (VM1) should read c(v+w)=cv+cw (not vw); and (VM2) and (VM3) should both read "For all c,d in F" (not in V). (Spotted by Brandon Peden.)
• On page 119, in the third displayed line, the summations over k should run from 1 to n, not m. (Spotted by Rippon Gupta.)
• On page 126, the term ceh in the formula for the 3 by 3 determinant should read cdh.
• Page 127, first displayed formula should be det(B) = det(A) + c det(C) = det(A). (Spotted by Will Funk.)
• Page 131, line 10: the exponent of omega should be -ij, not -j. (Spotted by William H. Millerd.)
• Page 135, axiom (MA0): m₁+m₁ should be m₁+m₂. (Spotted by Will Funk.)
• On page 139, top, I say I will consider right modules and, in the next definition, immediately contradict myself. It should of course say "right R-module" in the first line of the definition.
• Page 142, Theorem 5.6: "Suuppose" is su(u)pposed to be "Suppose". (Spotted by Csaba Szabó)
• Page 146, last line of Proposition 5.10: Ann(M₎ should read Ann(M₂).
• Page 153, first line: the second = in the displayed equation should be +.
• Page 156, middle: v_i=1 should read v_v+1.
• Page 170, line 9: "polynomials" mis-spelt.
• Page 170, second definition: should be "E a subfield of F".
• Page 172, first line of proof of Proposition 6.13: should be "Since c is algebraic ..."
• Page 180, last line: x³+x²-2 x+1 = 0. (not +0.) (Spotted by Karl Fedje.)
• Page 182, middle: lower-case g should be capital G.
• Page 184, the first displayed formula is garbled It should be: "(p^am choose p^a) is congruent to m mod p".
• Page 190, line 8: "A_n-1 intersect N = N" should read "A_n-1 intersect N = A_n-1". (Spotted by Matt Harvey)
• Page 205, line -14: opening bracket missing.
• Page 211, line -6: should be "psi(m) = phi(m)".
• Page 235, line -4: the second "the" should be "then". (Spotted by Karl Fedje.)
• Page 254, 2nd line of definition: f should be F; and 3rd line: F should be f. (Spotted by Richard Anderson.)
• Page 256, proof of Theorem 8.22: "Lemma 8.20" and "Lemma 8.21" should be "Proposition 8.20" and "Proposition 8.21". (Spotted by Csaba Szabó)
• Page 269: the reference to Stewart should be [25].
• Page 284, 4th last line of solution of 3.25: should be "smaller than x".
• Page 287, index entry for "associate class" should be page 49, not 48. (Spotted by Emily Ford.)
• Pages 288 and 294: the index entry for "Reed-Solomon code" should be 250-251, rather than 251. (Spotted by Karl Fedje.)
• Page 293: the index entry for "permutation group" should be page 87, not 81. (Spotted by Steve DiMauro.)

Special thanks to Robin Chapman, and to a number of students of Csaba Szabó, for spotting several misprints.

Please send me reports of further misprints.

Peter J. Cameron
13 April 2005