DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

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 complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$

Definition

Form the polynomial ring $R [x]$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution) and define the complex numbers by the quotient ring $C : = R [x] / (x^{2} + 1)$ (The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ). Write $a$ for the constant class $a + (x^{2} + 1)$ and set $i : = x + (x^{2} + 1) .$ Thus $i^{2} = - 1$ in the quotient. The constant-class map $R \to C$ is the specified real map; its injectivity and the field structure are proved in $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ↗, after which $C / R$ is a field extension in the sense of Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions.

Depends on

Dependency tree · 2 levels

Sources

T. Judson, Abstract Algebra: Theory and Applications, Extension Fields