On Galois Theory: an operational reformulation of the Galois framework within qdRIS

This note proposes an operator-based reformulation of the Galois framework using the Log Transfer Unitary Method (LTU), defined as the logarithm of the transfer function associated with a polynomial. Galois theory establishes the impossibility of expressing, in general, the roots of a polynomial of degree greater than or equal to five by means of elementary algebraic expressions. This limitation belongs strictly to the classical algebraic setting : it does not constrain functional representations, spectral reformulations, or dynamical approaches. By associating to a polynomial its transfer function and then applying LTU, one obtains a vectorial representation in which the action of the Galois group becomes an internal symmetry with no structural effect. The subsequent operator inversion transforms the roots into eigenmodes of a dynamical system, making the permutation of the roots irrelevant to the resulting structure. This reformulation does not contradict Galois theory : it shows instead that the combinatorial difficulty disappears once the problem is expressed in a different geometry. The note presents the steps of this reformulation and the structural invariances that ensure its coherence.