Tower of fields

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

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

A tower of fields may be finite or infinite.

Examples

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

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

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

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

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.