Constructive Resolution of Major Classes of Finite Groups as Galois Groups Over Q

We present a fully classical and constructive framework for realizing large classes
of finite groups as Galois groups over Q. Using embeddings into symmetric groups,
explicit polynomial constructions with discriminant control, auxiliary polynomials to
manage automorphisms, and rational specialization via Hilbert Irreducibility, we sys-
tematically realize:
• All finite solvable groups,
• All symmetric groups Sn and alternating groups An,
• Many finite simple groups via rigidity methods.
While the full Inverse Galois Problem remains open for certain large sporadic simple
groups, our method provides a structured constructive approach covering a vast portion of finite groups. All constructions occur strictly within classical Galois theory and explicit algebraic techniques.