ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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A quartic solved through its resolvent cubic

Example

For

$f (x) = x^{4} - 10 x^{2} + 1,$

the resolvent cubic is

$R_{f} (y) = y^{3} + 10 y^{2} - 4 y - 40 = (y - 2) (y + 2) (y + 10) .$

Taking the root $y = 2$ leads to

$x^{2} = 5 \pm 2 \sqrt{6},$

so the four roots are

$\pm (\sqrt{2} + \sqrt{3}), \pm (\sqrt{2} - \sqrt{3}) .$

Facts & Assumptions

Given: The quartic $f (x) = x^{4} - 10 x^{2} + 1$ .

[L1]

For $x^{4} + a x^{3} + b x^{2} + c x + d$ , the resolvent cubic is $y^{3} - b y^{2} + (a c - 4 d) y - (a^{2} d + c^{2} - 4 b d)$ (The coefficient formula and discriminant of the quartic resolvent).

[L2]

The roots of the resolvent are the three sums of products obtained from the three pairings of the four quartic roots (The resolvent cubic of a monic quartic).

Verification

technique · direct

1.1

Here $a = 0$ , $b = - 10$ , $c = 0$ , and $d = 1$ , so [L1] gives $R_{f} (y) = y^{3} + 10 y^{2} - 4 y - 40.$ Direct substitution shows that $y = 2$ is a root, and polynomial division yields the displayed factorization.

L1algebra

2.1

Use the resolvent root $y = 2$ to choose the pairing in [L2]. Seek a factorisation $f (x) = (x^{2} + u x + v) (x^{2} - u x + w) .$ The paired products give $v + w = y = 2$ , while the constant term gives $v w = 1$ , so $v = w = 1$ . Comparing the $x^{2}$ coefficient gives $v + w - u^{2} = - 10$ , hence $u^{2} = 12$ . Taking $u = 2 \sqrt{3}$ yields $f (x) = (x^{2} + 2 \sqrt{3} x + 1) (x^{2} - 2 \sqrt{3} x + 1) .$

L2step 1.1algebra

3.1

Solving the two quadratic factors in step 2.1 gives $- \sqrt{3} \pm \sqrt{2}$ and $\sqrt{3} \pm \sqrt{2}$ , which are exactly $\pm (\sqrt{2} + \sqrt{3}), \pm (\sqrt{2} - \sqrt{3}) .$ Thus the resolvent root leads to the four displayed quartic roots.

step 2.1algebra∎

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Sources

J. Ash, Basic Abstract Algebra, quartic examples (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, quartic resolvent examples (standard reference, not scraped)