This paper is the sixth part of the publication ``Selected chapters from algebra'', the first five having been published in previous issues of The Teaching of Mathematics, Vol\. I (1998), 1--22, Vol\. II, 1 (1999), 1--30, Vol\. II, 2 (1999), 65--80, Vol\. III, 1 (2000), 15--40, Vol\. III, 2 (2000), 63--82 and Vol\. IV, 1 (2001), 1--34. It is an English translation of: I. R. Shafarevich, {\it Izbranye glavy algebry. Glava VI. Beskonechnye mnozhestva}, Matematicheskoe obrazovanie, N~3--4(6--7), 1998, Moskva, str\. 2--36.