ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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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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$x^{2} + 1$ is irreducible over $R$ and split over $C$

Example

The polynomial $x^{2} + 1$ is irreducible in $R [x]$ , while over $C$ it factors as $x^{2} + 1 = (x - i) (x + i) .$

Facts & Assumptions

Given: The polynomial $f (x) = x^{2} + 1$ .

[L1]

The polynomial $x^{2} + 1$ is irreducible in $R [x]$ ( $x^{2} + 1$ is irreducible over $R$ ).

[L2]

Every nonconstant polynomial in $C [x]$ splits into linear factors (Every nonconstant polynomial in $C [x]$ splits into linear factors).

Verification

technique · direct

1.1L1

Fact [L1] gives the real-side claim immediately: $x^{2} + 1$ is irreducible over $R$ .

1.2L2algebra

In $C [x]$ one has $(x - i) (x + i) = x^{2} + 1,$ so $f$ has the two complex roots $i$ and $- i$ . This agrees with the general splitting statement [L2].

2.1step 1.1step 1.2∎

Thus $x^{2} + 1$ is the standard smallest-degree polynomial that is irreducible over $R$ but becomes reducible over $C$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, extension fields (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Chapter 5 (standard reference, not scraped)

x2+1x2+1 is irreducible over RR and split over CC