Here is a method for constructing a polynomial Q[x] with Galois group S n for

Here is a method for constructing a polynomial  ∫ ϵ Q[x] with Galois group S_n for a given n > 3. It depends on the fact that there exist irreducible polynomials of every degree in Z_p[x] (p prime; Corollary 5.9 below). First choose  ∫₁ ∫₂ ∫ ₃ ϵ Z[x] such that:

(i) deg  ∫₁ = n and ∫̅1 ϵ Z₂[x] is irreducible.

(ii) deg ∫₂= n and ∫̅₂ ϵ Z₃(x] factors in Z₃[.x-] as gh with g an irreducible of degree n - l and h linear;

(iii) deg  ∫₃ = n and ∫̅₃ ϵ Z₅[x] factors as gh or gh₁h₂ with g an irreducible quadratic in Z₅[x] and h,h₁,h₂ irreducible polynomials of odd degree in Z₅[x].

(a) Let ∫ = - 15∫₁ + lO∫₂ + 6∫₃ Then ∫ = ∫₁ (mod 2), ∫ = ∫₂ (mod 3), and ∫ = ∫₃ (mod 5). 

(b) The Galois group G of ∫ is transitive (since ∫̅ is irreducible in Z₂[x]).

( c) G contains a cycle of the type ζ = (i₁i₂• • • i_n-1) and element σ λ where σ is a transposition and λ a product of cycles of odd order. Therefore σ ϵ G, whence (i_ki_n) ϵ G for some k(l ≤ k ≤ n - 1) transitivity.

(d) G = S_n.

Data from Corollary 5.9

If K is a finite field and n ≥ 1 an integer, then there exists an irreducible polynomial of degree n in K[x].