Course 311: 2001–02. Supplementary Notes The Resolvent Cubic of a Quartic Polynomial

Let α, β, γ and δ be the roots of the quartic polynomial

x^{4} − px^{2} − qx − r,

so that

(x − α)(x − β)(x − γ)(x − δ) = x^{4} − px^{2} − qx − r.

Equating coefficients of x, we find that

α + β + γ + δ = 0,

and

p = −(αβ + αγ + αδ + βγ + βδ + γδ),
q = βγδ + αγδ + αβδ + αβγ,
r = −αβγδ.

Let

λ = (α + β)(γ + δ) = −(α + β)^{2} = −(γ + δ)^{2},
μ = (α + γ)(β + δ) = −(α + γ)^{2} = −(β + δ)^{2},
ν = (α + δ)(β + γ) = −(α + δ)^{2} = −(β + γ)^{2}.

We wish to express λ + μ + ν, μν + λν + λμ and λμν in terms of p, q and r.

To do this we eliminate α from the above expressions using the identity α + β + γ + δ = 0. We find

p = (β + γ + δ)(β + γ + δ) − γδ − βδ − βγ
= β^{2} + γ^{2} + δ^{2} + γδ + βδ + βγ,
q = βγδ − (β + γ + δ)(γδ + βδ + βγ)
= −(β^{2}γ + β^{2}δ + γ^{2}β + γ^{2}δ + δ^{2}β + δ^{2}γ) − 2βγδ,
r = β^{2}γδ + γ^{2}βδ + δ^{2}βγ.

Then

λ + μ + ν = −((γ + δ)^{2} + (β + δ)^{2} + (β + γ)^{2})
= −2(β^{2} + γ^{2} + δ^{2} + γδ + βδ + βγ)
= −2p,

λ^{2} + μ^{2} + ν^{2} = (γ + δ)^{4} + (β + δ)^{4} + (β + γ)^{4}
= γ^{4} + 4γ^{3}δ + 6γ^{2}δ^{2} + 4γδ^{3} + δ^{4}
+ β^{4} + 4β^{3}δ + 6β^{2}δ^{2} + 4βδ^{3} + δ^{4}
+ β^{4} + 4β^{3}γ + 6β^{2}γ^{2} + 4βγ^{3} + γ^{4}
= 2(β^{4} + γ^{4} + δ^{4}) + 4(β^{3}γ + β^{3}δ + γ^{3}β + γ^{3}δ + δ^{3}β + δ^{3}γ)
+ 6(γ^{2}δ^{2} + β^{2}δ^{2} + β^{2}γ^{2}),
p^{2} = β^{4} + γ^{4} + δ^{4} + 3(γ^{2}δ^{2} + β^{2}δ^{2} + β^{2}γ^{2})
+ 4(β^{2}γδ + γ^{2}βδ + δ^{2}βγ)
+ 2(β^{3}γ + β^{3}δ + γ^{3}β + γ^{3}δ + δ^{3}β + δ^{3}γ).

Therefore

λ^{2} + μ^{2} + ν^{2} = 2p^{2} − 8(β^{2}γδ + γ^{2}βδ + δ^{2}βγ)
= 2p^{2} − 8r.

But

4p^{2} = (λ + μ + ν)^{2} = λ^{2} + μ^{2} + ν^{2} + 2(μν + λν + λμ)

Therefore

μν + λν + λμ = 2p^{2} − \frac{1}{2}(λ^{2} + μ^{2} + ν^{2})
= p^{2} + 4r.

Finally, we note that

λμν = −((γ + δ)(β + δ)(β + γ))^{2}.

Now

(γ + δ)(β + δ)(β + γ) = β^{2}γ + β^{2}δ + γ^{2}β + γ^{2}δ + δ^{2}β + δ^{2}γ + 2βγδ
= −q.

Therefore

λμν = −(−q)^{2} = −q^{2}.

Thus λ, μ and ν are the roots of the resolvent cubic

x^{3} + 2px^{2} + (p^{2} + 4r)x + q^{2}.