In mathematics, specifically number theory, a tower of fields is a sequence of field extensions
A tower of fields may be finite or infinite.
Examples is a finite tower of the rational, real and complex numbers. The sequence obtained by setting and letting
for (i.e. is obtained from by adjoining a -th root of ) is an infinite tower. Similarly, if is a prime number, the pth cyclotomic tower of is obtained by setting and letting be the field obtained by adjoining to the -th roots of unity. This tower is of fundamental importance in Iwasawa theory. The Golod–Shafarevich theorem shows that there are infinite towers obtained by iterating the Hilbert class field construction to a number field.
References See section 4.1.4.