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Tower of fields

In mathematics, a sequence of field extensions

Summary

In mathematics, a sequence of field extensions

In mathematics, a tower of fields is a sequence of field extensions :F0 ⊆ F1 ⊆ ... ⊆ F**n ⊆ ... 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

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. F**n is obtained from F**n-1 by adjoining a 2nth root of 2), is an infinite tower.

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 {{Citation

Wikipedia Source

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