ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Long division, the Euclidean algorithm and a Bézout identity in $\mathbb Q[x]$

Example

In $\mathbb Q[x]$ , put $f=x^3-2x+1$ and $g=x^2-1$ . Long division and the Euclidean algorithm give

$f=xg+(-x+1),\qquad g=(-x-1)(-x+1).$

Thus $\gcd(f,g)=x-1$ and

$x-1=-f+xg.$

Facts & Assumptions

Given: The displayed polynomials $f,g\in\mathbb Q[x]$ .

[L1]

Polynomial division over a field gives a unique remainder of degree below the divisor (Division algorithm for polynomials over a field).

[L2]

The last nonzero Euclidean remainder, normalized to be monic, is the gcd and is a polynomial linear combination of the inputs (Bézout identity and the Euclidean algorithm for polynomials over a field).

[L3]

The rational numbers form a field (The rationals form a field).

Verification

technique · direct

1.1

Direct multiplication gives $x(x^2-1)+(-x+1)=x^3-2x+1=f$ , and $(-x-1)(-x+1)=x^2-1=g$ , so these are the divisions licensed by [L1] over the field [L3].

givenL1L3algebra

2.1

The last nonzero remainder is $-x+1$ , whose monic associate is $x-1$ ; negating the first remainder identity gives $x-1=-f+xg$ , in agreement with [L2].

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Thomas W. Judson, Abstract Algebra: Theory and Applications, Example 17.7 (standard reference, not scraped)
Neil Donaldson, Math 120B Notes, Section 23 (standard reference, not scraped)

Long division, the Euclidean algorithm and a Bézout identity in Q[x]\mathbb Q[x]Q[x]