# Can algebraic number theory be used to study L-rigs?

This question is a follow-up to Are there infinitely many L-rigs?, where I conjecture that every L-rig $$mathcal{L}$$ gives rise to an “L-field” $$mathcal {K}_{mathcal{L}}$$. Considering such an L-field as an extension of $$mathcal{K}_{mathcal{L}_{0}}cong mathbb{Q}$$, it should be possible to use the tools of algebraic number theory to study L-rigs.

In particular, can one study the decomposition of L-functions $$F$$ that are irreducible for the law $$otimes$$, and thus of prime degree $$d_{F}$$, into “L-ideals” of some “L-integers ring” $$mathcal{O}_{mathcal{K}_{mathcal{L}}}$$, this decomposition mimicking the decomposition of $$d_{F}$$ as an element of $$mathcal{O}_{mathbb{Q}}$$ into ideals of $$mathcal{O}_{mathbb{K}}$$ for some extension $$mathbb{K}$$ of $$mathbb{Q}$$?