Tower of fields

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

The name comes from such sequences often being written in the form

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

• 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