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

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

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

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

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

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

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

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

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.