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

[math]\displaystyle{ \begin{array}{c}\vdots \\ | \\ F_2 \\ | \\ F_1 \\ | \\F_0. \end{array} }[/math]

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

[math]\displaystyle{ F_{n+1}=F_n\left(2^{1/2^n}\right) }[/math]

(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

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