On the semi-simplicity conjecture for $\mathbb{Q}^{ab}$

TL;DR

This paper links the semi-simplicity conjecture for finitely generated fields to its validity for finite fields and the maximal abelian extension of rationals, suggesting a pathway to prove the conjecture more broadly.

Contribution

It demonstrates that the semi-simplicity conjecture for finitely generated fields can be derived from its cases for finite fields and the maximal abelian extension of .

Findings

Semi-simplicity conjecture for finitely generated fields depends on finite fields and ^{ab} cases.

Reduces the problem to known cases in specific fields.

Provides a conditional pathway to prove the conjecture more generally.

Abstract

We show that the semi-simplicity conjecture for finitely generated fields follows from the conjunction of the semi-simplicity conjecture for finite fields and for the maximal abelian extension of the field of rational numbers.