Here is a list of known misprints in the book Introduction to Algebra by Peter J. Cameron.

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, ..., an be the roots of the minimal polynomial of a over F. Now G permutes {a1, ..., an} transitively, and hence regularly. So the proper subgroup H cannot act transitively on this set. Let {a1, ..., ar} be an orbit. Then the coefficients of the monic polynomial with roots a1, ..., ar, 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:

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