Examples

• The sequence obtained by letting F₀ be the rational number
  Rational number
  In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
  
  s Q, and letting

|adjoining]] a 2ⁿth root

Nth root

In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

 of 2) is an infinite tower.

• If p is a prime number
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
   the pth cyclotomic tower of Q is obtained by letting F₀ = Q and F_n be the field obtained by adjoining to Q the pⁿth roots of unity
  Root of unity
  In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
  
  . This tower is of fundamental importance in Iwasawa theory
  Iwasawa theory
  In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...
  
  .
• The Golod–Shafarevich theorem
  Golod–Shafarevich theorem
  In mathematics, the Golod–Shafarevich theorem was proved in 1964 by two Russian mathematicians, Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which has consequences in various branches of algebra.-The inequality:...
  
   shows that there are infinite towers obtained by iterating the Hilbert class field
  Hilbert class field
  In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime...
  
   construction to a number field.