Samiksha Jaiswal (Editor)

Tower of fields

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

F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ...

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

⋮ | F 2 | F 1 | F 0 .

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
    • (i.e. Fn+1 is obtained from Fn 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 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.

    • References

    Tower of fields Wikipedia
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