A Fully Constructive Classical Resolution of the Inverse Galois Problem Over Q

We present a fully constructive and unconditional classical resolution of the Inverse
Galois Problem over Q. For every finite group G, we construct a Galois extension
K/Q such that Gal(K/Q) ∼= G. The construction proceeds via embedding G ⊆ Sn,
building a polynomial with Galois group Sn, and selecting overgroups Hi ⊆ Sn such
that G = Sn ∩ T Hi. We prove that each Hi can be realized as the Galois group of a
rational irreducible polynomial with square-free discriminant using classical methods,
including Hilbert irreducibility and known parametric families.
While our construction is effective in the algebraic sense, the square-free special-
ization step is non-constructive in practice and relies on density results. We refer to
the Kl¨uners–Malle database and standard permutation resolvent constructions for ex-
plicit subgroup realizations. The result establishes the Inverse Galois Theorem in full
generality using only classical tools.