{{Short description|In mathematics, a sequence of field extensions}} In mathematics, a '''tower of fields''' is a sequence of field extensions :{{nowrap|''F''<sub>0</sub> ⊆ ''F''<sub>1</sub> ⊆ ... ⊆ ''F''<sub>''n''</sub> ⊆ ...}} The name comes from such sequences often being written in the form :<math>\begin{array}{c}\vdots \\ | \\ F_2 \\ | \\ F_1 \\ | \\ \ F_0. \end{array}</math> A tower of fields may be finite or infinite.
==Examples== *{{nowrap|'''Q''' ⊆ '''R''' ⊆ '''C'''}} is a finite tower with rational, real and complex numbers. *The sequence obtained by letting ''F''<sub>0</sub> be the rational numbers '''Q''', and letting ::<math>F_{n} = F_{n-1}\!\left(2^{1/2^n}\right), \quad \text{for}\ n \geq 1</math> :(i.e. ''F''<sub>''n''</sub> is obtained from ''F''<sub>''n-1''</sub> by adjoining a 2<sup>''n''</sup>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 ''F''<sub>0</sub> = '''Q''' and ''F''<sub>''n''</sub> be the field obtained by adjoining to '''Q''' the ''p<sup>n</sup>''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== *Section 4.1.4 of {{Citation | last=Escofier | first=Jean-Pierre | title=Galois theory | publisher=Springer-Verlag | series=Graduate Texts in Mathematics | volume=204 | year=2001 | isbn=978-0-387-98765-1 }}
Category:Field extensions