Symmetric groups as Galois groups over the rationals

For every positive degree n, there is a monic integral polynomial of degree n, irreducible over ℚ, with full symmetric Galois group. For n ≥ 2, choose an irreducible reduction modulo 2, factor degrees (1, n - 1) modulo 3, and exactly one quadratic factor with all other degrees odd modulo 5. The coefficientwise Chinese remainder theorem produces one polynomial with all three reductions. The reduction criterion then gives every permutation of its roots. Degree one is witnessed by X.

The construction uses the finite-field factorization patterns and monic Chinese remainder lifting already provided by Tau Ceti, and the criterion TauCeti.surjective_galActionHom_of_factorDegrees.

References

• B. L. van der Waerden, Algebra I, §61.
• J.-P. Serre, Topics in Galois Theory, second edition, §4.4.

theorem TauCeti.exists_monic_int_polynomial_hasFullSymmetricGaloisGroup (n : ℕ) (hn : 1 ≤ n) : ∃ (f : Polynomial ℤ), f.Monic ∧ f.natDegree = n ∧ Irreducible (Polynomial.map (Int.castRingHom ℚ) f) ∧ (Polynomial.map (Int.castRingHom ℚ) f).HasFullSymmetricGaloisGroup

Every positive degree occurs for a monic integral polynomial irreducible over ℚ with full symmetric Galois group.