Tower of fields
- F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ...
The name comes from such sequences often being written in the form
A tower of fields may be finite or infinite.
- 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
- If p is a prime number the p th cyclotomic tower of Q is obtained by letting F0 = Q and Fn 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.