Tower of fields

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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

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

(i.e. F_n+1 is obtained from F_n by adjoining a 2ⁿth root of 2) is an infinite tower.

If p is a prime number the pth cyclotomic tower of Q is obtained by letting F₀ = Q and F_n be the field obtained by adjoining to Q the pⁿth 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.

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

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