# Tower of fields

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

*F*_{0}⊆*F*_{1}⊆ ... ⊆*F*_{n}⊆ ...

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

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
*F*_{0}be the rational numbers**Q**, and letting

- If
*p*is a prime number theof*p*th cyclotomic tower**Q**is obtained by letting*F*_{0}=**Q**and*F*_{n}be the field obtained by adjoining to**Q**the*p*th roots of unity. This tower is of fundamental importance in Iwasawa theory.^{n} - 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 {{#invoke:citation/CS1|citation

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