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
![{\displaystyle {\begin{array}{c}\vdots \\|\\F_{2}\\|\\F_{1}\\|\\\ F_{0}.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7936cd531f7a83adff8c6aa3089f7dfd088193)
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
![{\displaystyle F_{n}=F_{n-1}\!\left(2^{1/2^{n}}\right),\quad {\text{for}}\ n\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aee533f74c5d2f8718b3b5282d745ce743a9d5a1)
- (i.e. Fn is obtained from Fn-1 by adjoining a 2n th root of 2), is an infinite tower.
References