Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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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FALSE: the real numbers are algebraically closed

Statement

False claim: the field R is algebraically closed.

Facts & Assumptions

Given: The polynomial f(x)=x2+1∈R[x].

[L1]

A field is algebraically closed exactly when every nonconstant polynomial over it has a root in the field (An algebraically closed field: every nonconstant polynomial has a root in the field).

[L2]

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

Refutation

technique · direct
1.1

By [L2], the polynomial x2+1 has no real root.

L2algebra
2.1

Therefore R fails the defining property in [L1] for algebraic closedness.

L1step 1.1
3.1

So the real numbers are not algebraically closed.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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