What the work claims
A polynomial equation is solvable by radicals if and only if its Galois group is solvable. A reduced quadratic surd and the negative reciprocal of its conjugate have purely periodic continued fractions whose repeating blocks are mirror images. Finite fields and normal subgroups were introduced as foundational algebraic objects.
How it was done
Galois related solvability of polynomial equations to the structure of a group of permutations of their roots — the Galois group — and introduced proper decomposition of groups, leading to the definition of normal subgroups.
What holds up
His necessary and sufficient condition for solvability by radicals holds. His characterisation of solvable groups via nested normal subgroups with abelian quotients holds. His mirror-image theorem for continued fractions of reduced quadratic surds holds. His introduction of finite fields and normal subgroups holds.
What does not
The sources do not report any experimental validation, numerical computation, or application to physical systems. Nothing is said about computational complexity, algorithmic implementation, or extension beyond polynomials and quadratic surds.
Why it matters beyond the lab
These ideas define the boundary between what can and cannot be computed algebraically — a distinction that shapes modern computer algebra, error-correcting codes, and cryptographic protocols built on finite fields.
Is it worth your time
Yes. His criteria remain the definitive test for radical solvability, and his definitions underpin modern algebra, cryptography, and coding theory.
