Tower of fields

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (September 2026) (Learn how and when to remove this message)

In mathematics, specifically number theory, a tower of fields is a sequence of field extensions

${\displaystyle F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{n}\subseteq \cdots }$

A tower of fields may be finite or infinite.

Examples

• ${\displaystyle \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$ is a finite tower of the rational, real and complex numbers.
• The sequence obtained by setting ${\displaystyle F_{0}=\mathbb {Q} }$ and letting

${\displaystyle F_{n}=F_{n-1}{\big (}2^{1/2^{n}}{\big )}}$

for ${\displaystyle n\geq 1}$ (i.e. ${\displaystyle F_{n}}$ is obtained from ${\displaystyle F_{n-1}}$ by adjoining a ${\displaystyle 2^{n}}$-th root of ${\displaystyle 2}$) is an infinite tower.

• Similarly, if ${\displaystyle p}$ is a prime number, the pth cyclotomic tower of ${\displaystyle \mathbb {Q} }$ is obtained by setting ${\displaystyle F_{0}=\mathbb {Q} }$ and letting ${\displaystyle F_{n}}$ be the field obtained by adjoining to ${\displaystyle \mathbb {Q} }$ the ${\displaystyle p^{n}}$-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.

References

• Escofier, Jean-Pierre (2001) [1997]. Galois Theory. Graduate Texts in Mathematics. Vol. 204. Translated by Schneps, Leila (2nd ed.). New York: Springer. ISBN 978-1-4612-6558-0. See section 4.1.4.

This number theory–related article is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e