Infinite galois theory

Abstract

This chapter discusses the infinite Galois theory. The main problem of Galois theory is to find out whether or not each finite group occurs as a Galois group over the field ℚ of rational numbers. Cyclotomic extensions supply all finite abelian groups as Galois groups over ℚ. The Hilbert irreducibility theorem combined with the Riemann existence theorem provides many nonabelian simple groups and quasi simple groups. Infinite Galois theory extends the question about the structure of G(ℚ) to a question about the structure of absolute Galois groups of other distinguished fields. The maximal prosolvable quotient of G(ℚ ab ) is the free prosolvable group on countably many generators. The theory of profinite groups is an outcome of infinite Galois theory. As for finite groups, each profinite group occurs as a Galois group of some Galois extension. The inverse problem of infinite Galois theory is to characterize those profinite groups that occur as absolute Galois groups of fields.