Insolvability of quintics.

Somebody who thinks logically is a nice contrast to the real world or, alternatively, common sense is the least common of the senses, it is not so common.

 

Theorem. Let f be an irreducible polynomial of degree 5 over a subfield F of the complex numbers (F ⊆ ℂ), whose Galois group is either the alternating group A₅ or the symmetric group S₅, then f is not solvable. Therefore, there are quintics that are not solvable.

Proof.

For the sake of contradiction, suppose f is solvable ↭ Gal(f) is solvable ⊥ S₅ or A₅ are not solvable.

Theorem. Let f be an irreducible polynomial of degree 5 over a subfield F of the complex numbers (F ⊆ ℂ), whose Galois group is either the alternating group A₅ or the symmetric group S₅, then f is not solvable.

Proof. (2nd version. Based on Algebra, Second Edition by Michael Artin, Theorem 16.12.4, pg 503)

Let G = Gal(f). First, let’s suppose G = S₅, and K be the splitting field of f over F, so G = Gal(K/F) = Gal(f), D = Disc(f), δ = $\sqrt{D}$ ∈ K.

By assumption, G = S₅ ⇒ G = Gal(f) ⊄ A₅ ⇒ [Recall If δ ∈ F (Δ is a square in F), then G ⊆ An.] δ ∉ F ⇒ [F(δ) : F] = 2 ⇒ [$A_n$ is the only subgroup of $S_n$ of index 2] Gal(K/F(δ)) = A₅, so the Galois group of f over F(δ) is A₅. Therefore, if we demonstrate that if Gal(f) = A₅, then f is not solvable, we are done (because f is not solvable over F(δ) ⇒ f is not solvable over F)

Therefore, we may assume that G is the alternating group A₅, a simple group, f ∈ F[x], deg(f) = 5, Gal(f) = A₅.

For the sake of contradiction, let’s suppose f it is solvable over F. Let α ∈ K be a root of f, then α is solvable over F, (Recall: solvable ↭ tower of Abelian fields ↭ tower of cyclic fields ↭ tower of cyclic fields of prime order) then there exist a tower of extensions which are cyclic of prime degree.

F = F₀ ⊆ F₁ ⊆ F₂ ⊆ ··· ⊆ F_r, α ∈ F_r, F_i/F_i-1 is Galois, [F_i : F_i-1 is prime]

Lemma. Let F’/F be a Galois extension of prime degree p. Let K and K' be the splitting field of f over F and F' respectively. Then, Gal(K'/F')≋A₅.

Proof.

K is the splitting field of f (irreducible) over F, α₁, α₂, α₃, α₄, α₅ are five distinct elements, K = F(α₁, α₂, α₃, α₄, α₅). K’ is the splitting field of f over F'.

[F’: F] = p, Gal(F’/F) ≋ ℤ/pℤ, and it cannot have proper intermediate fields because it is a prime number.

By assumption, F’/F is Galois ⇒ F’ is the splitting field of a polynomial g ∈ F[x] ⇒ g is irreducible. For the sake of contradiction, let’s suppose that g is reducible, say g = h·h’, deg(h)>0, deg(h’)>0. However, we can assume that deg(h)>1, deg(h’)>1, and (h, h’) = 1. Otherwise, g = h(x-a), a ∈ F and the splitting field of h is the same as the splitting field of g

⊥ Therefore, g is irreducible of deg p, F’ = F(β₁, ···, β_p). K’ = F(α₁, α₂, α₃, α₄, α₅, β₁, ···, β_p) is the composite of F’ and K in $\bar F$, and notice that K’ is the splitting field of f over F’ and it is generated by its roots, i.e., K’ = F’(α₁, α₂, α₃, α₄, α₅)

Each of the extension fields is a Galois extension, and the Galois groups have been labeled in the diagram, e.g., K’/F is Galois because it is the composite of K/F and F’/F, and they are both Galois extensions (K’ is the splitting field of fg over F).

Let N = Gal(K’/F), N is Galois. From Galois’ fundamental theorem we know H ◁ N and N/H ≋ G’ ≋ ℤ/pℤ (F’/F is Galois), H’ ◁ N and N/H’ ≋ G ≋ A₅ (K’/F is Galois).

Next, the reader should notice that H ∩ H’ = {id} because σ ∈ H’ = Gal(K’/K), σ is an F-automorphism of K’ that fixes the roots α₁, ···, α₅.

σ ∈ H = Gal(K’/F’), σ is an F-automorphism of K’ that fixes the root β₁, ···, β_p. Therefore, σ ∈ H ∩ H’ ⇒ σ is an F-automorphism of K’ that fixes β₁, ···, β_p, α₁, ···, α₅, but K’ = F(α₁, α₂, α₃, α₄, α₅, β₁, ···, β_p) ⇒ σ = id, H ∩ H’ is the trivial group.

Consider the canonical map, Φ: N → N/H ≋ G’. Let’s restrict the canonical map to the subgroup H’ (H’◁ N), Φ|_H’: H’ → N/H. The Kernel of this restriction is the trivial group: Ker(Φ|_H’) = Ker(Φ) ∩ H’= H ∩ H’ = {id} ⇒ Φ|_H’: H’ → N/H ≋ G’ ≋ ℤ/pℤ, the restriction Φ|_H’ is injective ⇒ It maps H’ isomorphically to a subgroup of G’, a cyclic group of prime order, ℤ/pℤ, so there are only two options: either H’ is the trivial group, or else H’ is cyclic of order p.

1. H’ is the trivial group, H’ = {id}, N → N/H’≋ G ≋ A₅, H’◁ N ⇒ so the map is onto (H’◁ N, then there is always a canonical surjective group homomorphism from N to the quotient group N/H’ that sends an element g∈ N to the coset determined by g), but it is also 1-1 (its kernel is H’ = [By assumption] {id}) ⇒ N ≋ A₅ ⇒ [A₅ is simple] N is simple.

   N does have a normal subgroup, namely H ◁ N, such that N/H ≋ G’ ≋ ℤ/pℤ, N ≋ A₅ ⇒ |N| = 60, N/H ≋ G’ ≋ ℤ/pℤ ⇒ |N/H| = p ⇒ |H| = 60/p ⇒ |H| ≠ 1 and |H| ≠ 60, so H is a proper, non-trivial normal subgroup of N, but N is supposed to be simple ⊥

2. H’ ≋ ℤ/pℤ, |N| = [N/H ≋ G’ ≋ ℤ/pℤ] = |G’||H| = |H|p. |N| = [N/H’≋G ≋ A5] = |G||H’| = 60p. Therefore, |H| = 60.

Consider the canonical map ψ:N → N/H’ ≋ G (it is always surjective), and restrict it to the subgroup H, ψ|_H.

The kernel of this restriction is the trivial group: Ker(ψ|_H) = Ker(ψ) ∩ H = H’ ∩ H = {id}, so ψ|_H is injective. Therefore, H is isomorphic to a subgroup of G. However, |H| = 60 = [G ≋ A5] |G| ⇒ Since both groups have order 60, the restriction ψ|_H is indeed an isomorphism ⇒ H ≋ G ≋ A5∎

[Continuing with the proof…]

Notice:

1. By assumption K is the splitting field of f over F. Gal(K/F) = Gal(f) =A₅.
2. By the previous result, let K₁ be the splitting field of f over F₁ ⇒ Gal(K₁/F₁) = A₅
3. We repeat the process up to K_r, the splitting field of f over F_r, Gal(K_r/F_r) = A₅

However, f is not irreducible in F_r[x] because it has a root, namely α ∈ F_r ⇒ f = (x - α)f’, f’ ∈F_r[x], deg(f’) = 4

Since K_r is the splitting field of f over F_r ⇒ K_r is the splitting field of f’ over F_r, deg(f’) = 4 ⇒ Gal(K_r/F_r) ≤ S₄, and in particular |Gal(K_r/F_r| ≤ |S₄| = 24, but we have already established that Gal(K_r/F_r) = A₅, so has order 60 ⊥ Therefore, f is not solvable over F.