Tower of fields

In mathematics, a tower of fields is a sequence of field extensions
 * F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ...

The name comes from such sequences often being written in the form
 * $$\begin{array}{c}\vdots \\ | \\ F_2 \\ | \\ F_1 \\ | \\ \ F_0. \end{array}$$

A tower of fields may be finite or infinite.

Examples

 * Q ⊆ R ⊆ C is a finite tower with rational, real and complex numbers.
 * The sequence obtained by letting F0 be the rational numbers Q, and letting
 * $$F_{n} = F_{n-1}\!\left(2^{1/2^n}\right), \quad \text{for}\ n \geq 1$$
 * (i.e. Fn is obtained from Fn-1 by adjoining a 2n&hairsp;th root of 2), is an infinite tower.


 * If p is a prime number the p&hairsp;th cyclotomic tower of Q is obtained by letting F0 = Q and Fn be the field obtained by adjoining to Q the pn&hairsp;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.