ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Example

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

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

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

$x - 1 = - f + x g .$

Facts & Assumptions

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

[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 $x (x^{2} - 1) + (- x + 1) = x^{3} - 2 x + 1 = f$ , 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 $x - 1 = - f + x g$ , in agreement with [L2].

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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

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]Q[x]