Tower of fields
# Tower of fields

In mathematics, a tower of fields is a sequence of field extensions

    $F_0$ ⊆ $F_1$ ⊆ ... ⊆ $F_n$ ⊆ ...

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 $F_0$ be the rational numbers Q, and letting

       $F_{n+1}=F_{n}\!\left(2^{1/2^{n}}\right)$
    (i.e. $F_{n+1}$ is obtained from $F_n$ by adjoining a $2^n$ th root of 2) is an infinite tower.
* If p is a prime number the p th cyclotomic tower of Q is obtained by letting $F_0$ = Q and $F_n$ be the field obtained by adjoining to Q the $p^n$ 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.
