Tower of fields

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

F₀ ⊆ F₁ ⊆ ... ⊆ 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 F₀ be the rational numbers Q, and letting

$F_{n}=F_{n-1}\!\left(2^{1/2^{n}}\right),\quad {\text{for}}\ n\geq 1$

(i.e. F_n is obtained from F_n-1 by adjoining a 2ⁿ 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 F₀ = Q and F_n be the field obtained by adjoining to Q the pⁿ 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

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