Tower of fields

In mathematics, a sequence of field extensions

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

F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ...

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

⋮ | F 2 | F 1 |   F 0 . {\displaystyle {\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 ( 2 1 / 2 n ) , for   n ≥ 1 {\displaystyle F_{n}=F_{n-1}\!\left(2^{1/2^{n}}\right),\quad {\text{for}}\ n\geq 1}
(i.e. Fn is obtained from Fn-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 Fn 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