Monomorphism_from_Rationals_to_Ordered_Field

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $\struct {F, +, \circ, \le}$ be a totally ordered field. There is one and only one (ring) monomorphism from the totally ordered field $\Q$ onto $F$. Its image is the prime subfield of $F$.

Why this node was retired

The posted statement is

theorem Monomorphism_from_Rationals_to_Ordered_Field {F : Type _} [Field F] [LinearOrder F] : ∃ f : ℚ →+* F, Function.Injective f := by sorry

Field and LinearOrder structures are independent; finite positive-characteristic fields qualify.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Use a field with a compatible ordered-ring structure, or simply a characteristic-zero field, and assert the canonical ℚ-algebra map is the unique unital ring homomorphism ℚ→F and is injective. Its image is the prime subfield, not generally all F.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.