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

$\begin{array}{c}⋮\\ |\\ F_2\\ |\\ F_1\\ |\\ F_0.\end{array}$

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

$F_n=F_{n-1}\left(2^{1/2^n}\right),\text{for}n\ge 1$

(i.e. F_n is obtained from F_n-1 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.

References

• Section 4.1.4 of