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*=*a*_{1}, ...,
*a _{n}* be the roots of the minimal polynomial of

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):
*g*^{1}should be*g*_{1}. (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
((
*Ng*_{1})(*Ng*_{2}))(*Ng*_{3}), not ((*Ng*_{1})(*Ng*_{2}))(*Ng*_{2}). (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*× log_{2}*n*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):
*m*_{1}+*m*_{1}should be*m*_{1}+*m*_{2}. (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(_{)}*M*)._{2} - Page 153, first line: the second = in the displayed equation should be +.
- Page 156, middle:
*v*should read_{i=1}*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:
*x*^{3}+*x*^{2}-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*is congruent to^{a}m choose p^{a})*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