Tower of fields explained

In mathematics, a tower of fields is a sequence of field extensions

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

• 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), for n \geq 1

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