Galois Theory, 4th Edition

282 Algebraically Closed Fields

irreducible polynomial over K(i) has degree 1, otherwise a splitting field would have

finite degree > 1 over K(i). Therefore K(i) is algebraically closed.

Corollary 23.13 (Fundamental Theorem of Algebra). The field C of complex

numbers is algebraically closed.

Proof. Put R = K in Theorem 23.12 and use Lemma 23.3.

EXERCISES

23.1 Show that A

5

has no subgroup of order 15.

23.2 Show that a subgroup or a quotient of a p-group is again a p-group. Show that

an extension of a p-group by a p-group is a p-group.

23.3 Show that S

n

has trivial centre if n ≥3.

23.4 Prove that every group of order p

2

(with p prime) is abelian. Hence show that ...