# Tower of fields

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

F0F1 ⊆ ... ⊆ Fn ⊆ ...

The name comes from such sequences often being written in the form

${\displaystyle {\begin{array}{c}\vdots \\|\\F_{2}\\|\\F_{1}\\|\\F_{0}.\end{array}}}$

A tower of fields may be finite or infinite.

## Examples

• QRC is a finite tower with rational, real and complex numbers.
• The sequence obtained by letting F0 be the rational numbers Q, and letting
${\displaystyle F_{n+1}=F_{n}\left(2^{1/2^{n}}\right)}$
(i.e. Fn+1 is obtained from Fn by adjoining a 2nth root of 2) is an infinite tower.

## References

• Section 4.1.4 of Escofier, Jean-Pierre (2001), Galois theory, Graduate Texts in Mathematics, 204, Springer-Verlag, ISBN 978-0-387-98765-1