Tower of fields

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

F_{n+1}=F_{n}\left(2^{1/2^{n}}\right)

(i.e. F_n+1 is obtained from F_n by adjoining a 2ⁿth 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 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.

gollark: Unfortunately, my excellent code appears to not work properly.```c#include <stdio.h>#include <signal.h>#include <string.h>#include <stdlib.h>#include <sys/mman.h>#include <unistd.h>static void handler(int sig, siginfo_t *info, void *utterly_worthless_argument) { printf("oh bees segfault %08x\n", info->si_addr); int ps = getpagesize(); long ad = (long)info->si_addr; ad = ad - (ad % ps); mmap((void*)ad, 0x10000, PROT_NONE, MAP_ANONYMOUS|MAP_FIXED, -1, 0);}int main() { struct sigaction sa; sigemptyset(&sa.sa_mask); sa.sa_flags = SA_NODEFER; sa.sa_sigaction = handler; sigaction(SIGSEGV, &sa, NULL); *(int*)NULL = -3; printf("thing done\n"); return 0;}```

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References

Section 4.1.4 of Escofier, Jean-Pierre (2001), Galois theory, Graduate Texts in Mathematics, 204, Springer-Verlag, ISBN 978-0-387-98765-1

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