##  [Infinite Class Field Tower](/infinite-class-field-tower-0) 

 Definition

A tower of number fields obtained by iteratively taking the Hilbert class field (maximal unramified abelian extension) of the previous level that never stabilizes — i.e., the sequence of class fields is infinite and class numbers do not drop to one uniformly, indicating failure of termination of the class field tower.

 

 

 

 

 

 





## Principle

Principle

Iterate unramified abelian extension formation: start with a number field, pass to its Hilbert class field, then repeat; an infinite tower arises when this process produces infinitely many distinct layers, reflecting persistent nontrivial class groups at every stage.

 

 

 

 

 





## Demonstration

Demonstration

Begin with a base number field with sufficiently large class group; compute its Hilbert class field to obtain a new field whose class group can again be nontrivial, and continue; explicit constructions or criteria (e.g., Golod–Shafarevich type inequalities) produce families where infinitely many successive Hilbert class fields exist.

 

 

 

 

## Misapplication

Misapplication

Assuming every number field admits a finite class field tower and using that to deduce boundedness of class numbers universally; conversely, misreading finite stabilization at low levels as proof of eventual termination without checking deeper arithmetic obstructions is incorrect.

 

 

 

 

 





## Consequence

Consequence

Existence of an infinite class field tower exhibits profound failure of class field tower termination and implies unbounded growth of unramified abelian extensions; it influences understanding of class groups, Galois group structure of maximal unramified extensions, and heuristics about discriminants.

 

 

 

 

## Reversal

Reversal

A finite class field tower (stabilization after finitely many steps) is the opposite phenomenon: eventually the Hilbert class field is trivial and no further nontrivial unramified abelian extensions exist, indicating eventual vanishing of further class group growth.

 

 

 

 

 





## Boundary

Boundary

A phenomenon about unramified abelian extensions of number fields; it does not concern ramified extensions or arbitrary nonabelian unramified extensions unless specifically extended, and its detection often needs cohomological or pro‑p group analytic tools rather than elementary class number checks.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension appears between heuristics that predict generic finiteness of class field towers and explicit examples or criteria showing infinitude; nearby concepts include growth of p‑class groups or nonabelian unramified extensions, which are related but not identical phenomena.

 

 

 

 

 





## Synthesis

Synthesis

An infinite class field tower is the arithmetic situation where repeated passage to Hilbert class fields never terminates: iterate Hilbert class field formation, observe persistent nontrivial class groups at each level, and thereby obtain an infinite ascending chain of unramified abelian extensions.