Tower of fields

Short description: Sequence of field extensions in mathematics

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

• $\mathbb{\mathbb{Q}}\subseteq \mathbb{ℝ}\subseteq \mathbb{ℂ}$ is a finite tower of the rational, real and complex numbers.
• The sequence obtained by setting $F_0=\mathbb{\mathbb{Q}}$ and letting

$F_n=F_{n-1}\left(2^{1/2^n}\right)$

for $n\ge 1$ (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 $\mathbb{\mathbb{Q}}$ is obtained by setting $F_0=\mathbb{\mathbb{Q}}$ and letting $F_n$ be the field obtained by adjoining to $\mathbb{\mathbb{Q}}$ 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). Galois Theory. Graduate Texts in Mathematics. 204 (2nd ed.). New York: Springer. ISBN 978-1-4612-6558-0.  See section 4.1.4.