Can a chain of number systems keep growing forever without getting more tangled per step, and how does it manage that?

This explores why Golod–Shafarevich class field towers give fields of ever-larger degree whose root discriminant never grows. The corpus has no material on this: it's a library about AI and LLM research, and none of the twelve retrieved notes deals with number fields, class groups or discriminants. The matches came from surface vocabulary like 'towers', 'bounded' and 'growth'. Here is a short orientation from general background, not from the collection.

The mechanism is simple once you see it. The discriminant of a number field measures ramification, which is roughly how primes 'collapse' when you extend the field. For a tower K ⊂ L, the discriminants multiply: |d_L| = |d_K|^[L:K] times the norm of the relative discriminant. If the extension L/K is unramified, that relative discriminant is trivial, so |d_L|^(1/[L:Q]) = |d_K|^(1/[K:Q]). In other words, the root discriminant stays exactly the same. A class field tower is built entirely out of unramified extensions: each step is the maximal unramified abelian p-extension (the p-Hilbert class field) of the field before it. Every field in the tower therefore has the same root discriminant as the base field. Golod and Shafarevich showed that if the p-part of the base field's class group has enough generators compared with its unit rank, the tower never stops. That gives infinitely many fields of growing degree with one fixed root discriminant. This answered a question about root discriminants going to infinity with the degree in the negative, and it is why Odlyzko's lower bounds on root discriminants are finite constants rather than bounds that grow with the degree. Martinet's explicit example has a root discriminant of about 92.4. Narrowing the gap between such examples and Odlyzko's bounds is still open.

The nearest the corpus comes is the meeting point of machine learning and pure mathematics. Can opaque machine learning models help prove new mathematics? argues that opaque models can help produce rigorous results when an external checker, such as a proof assistant or numerical analysis, confirms what they output. Searching for small-discriminant fields or infinite towers is the kind of large computational hunt where that pairing could matter. That connection is speculative, though, and the corpus doesn't make it. If this is the line you want to follow, look in number theory references such as Cassels–Fröhlich, Roquette's chapter on class field towers, or Martinet's and Odlyzko's papers, not in this library.