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Tower of fields

Summary

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 edit

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

(i.e. F_n+1 is obtained from F_n 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 edit

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