Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13
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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.

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

x2+1 is irreducible over R

Statement

The polynomial x2+1 is irreducible in R[x].

Facts & Assumptions

Given: The polynomial x2+1∈R[x].

[F1]

The real numbers form an ordered field (The reals form a totally ordered field).

[F2]

In an ordered field, the square of every nonzero element is positive (Squares of nonzero elements are positive).

[F3]

A polynomial of degree two or three over a field is irreducible if and only if it has no root in that field (A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field).

Proof

technique · direct
1.1

For r∈R, either r=0, so r2=0, or [F2] gives r2>0. Thus r2+1>0 by the ordered-field laws in [F1].

F1F2
2.1

Hence x2+1 has no real root.

step 1.1
3.1

Since it has degree two, [F3] now gives its irreducibility over R.

F3step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 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