ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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$x^{4} - 10 x^{2} + 1$ has Galois group $V_{4}$ over $Q$

Example

The polynomial $x^{4} - 10 x^{2} + 1$ has Galois group $V_{4}$ over $Q$ .

Facts & Assumptions

Given: The quartic resolvent formula (The coefficient formula and discriminant of the quartic resolvent) and the fully split resolvent row of The five-case resolvent classification of an irreducible quartic Galois group.

[L1]

$Q (\sqrt{2}, \sqrt{3}) / Q$ is Galois with group $V_{4}$ (The complete Galois correspondence for $Q (\sqrt{2}, \sqrt{3}) / Q$ ).

Verification

technique · direct

1.1

Put $α = \sqrt{2} + \sqrt{3}$ . Then $(α^{2} - 5)^{2} = 24$ , so $α^{4} - 10 α^{2} + 1 = 0$ . Since $α^{- 1} = \sqrt{3} - \sqrt{2}$ , one recovers $\sqrt{2} = (α - α^{- 1}) / 2$ and $\sqrt{3} = (α + α^{- 1}) / 2$ , so $Q (α) = Q (\sqrt{2}, \sqrt{3})$ and the polynomial is the degree-four minimal polynomial of $α$ .

algebra

2.1

Its four distinct conjugates are $\sqrt{2} + \sqrt{3}$ , $\sqrt{2} - \sqrt{3}$ , $- \sqrt{2} + \sqrt{3}$ , and $- \sqrt{2} - \sqrt{3}$ , all in the biquadratic field. Thus this is the splitting field and [L1] gives Galois group $V_{4}$ .

step 1.1L1

3.1

The resolvent is $y^{3} + 10 y^{2} - 4 y - 40 = (y + 10) (y - 2) (y + 2)$ , so it splits completely over $Q$ , agreeing with the $V_{4}$ row.

step 2.1givenalgebra∎

Depends on

Used by

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Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, The Galois Correspondence, biquadratic examples (standard reference, not scraped)
K. Conrad, Galois Groups of Cubics and Quartics, Section 3 (standard reference, not scraped)

x4−10x2+1x4−10x2+1 has Galois group V4V4​ over QQ