ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-16

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.

The relative algebraic closure of $R$ in $C$ is all of $C$

Example

The relative algebraic closure of $R$ in $C$ is $C$ itself.

Facts & Assumptions

Given: A complex number $z = a + b i$ with $a, b \in R$ .

[L1]

The relative algebraic closure consists of the elements algebraic over the base field (The relative algebraic closure of $F$ in an extension $K$ ).

[L2]

Every complex number has a unique form $a + b i$ , with $i^{2} = - 1$ ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ).

Verification

technique · direct

1.1givenL2algebra

The nonzero real polynomial $(t - a)^{2} + b^{2}$ vanishes at $z$ , because $(z - a)^{2} + b^{2} = (b i)^{2} + b^{2} = 0$ .

2.1step 1.1L1∎

Thus every $z \in C$ is algebraic over $R$ , so [L1] identifies the relative algebraic closure with all of $C$ .

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

A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 1 (standard reference, not scraped)

The relative algebraic closure of RR in CC is all of CC