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
.
{\displaystyle {\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
-
F
n
=
F
n
−
1
(
2
1
/
2
n
)
,
for
n
≥
1
{\displaystyle F_{n}=F_{n-1}\!\left(2^{1/2^{n}}\right),\quad {\text{for}}\ n\geq 1}
-
F
n
=
F
n
−
1
(
2
1
/
2
n
)
,
for
n
≥
1
{\displaystyle F_{n}=F_{n-1}\!\left(2^{1/2^{n}}\right),\quad {\text{for}}\ n\geq 1}
- (i.e. Fn is obtained from Fn-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 Fn 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 Escofier, Jean-Pierre (2001), Galois theory, Graduate Texts in Mathematics, vol. 204, Springer-Verlag, ISBN 978-0-387-98765-1