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Canonical static concept record

Polynomial-quotient complex field

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Summary

The quotient ring $\mathbb R[x]/(x^2+1)$ , which is a field isomorphic to $\mathbb C$ .

Record metadata

Canonical name: Polynomial-quotient complex field
Concept ID: complex_polynomial_quotient
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis, Algebra, Set theory, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Quotient realization

Carrier(s)

quotient of $\mathbb R[x]$

Data

residue class $i=[x]$

Axioms / constraints

$i^2=-1$

Notes

Because $x^2+1$ is irreducible over $\mathbb R$ , the quotient is a field.

Concept sources

Wikipedia — Complex number — Complex number · encyclopedic reference · source ID wp-complex-number
nLab — complex number — complex number · mathematical reference · source ID nlab-complex-number

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Real numbers $\mathbb R$ Polynomial-quotient complex field

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: real_numbers_to_complex_quotient
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: adjoin a root of $x^2+1$

Authored explanation

Form $\mathbb R[x]/(x^2+1)$ , in which the class of $x$ squares to $-1$ .

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Complex number — Complex number · encyclopedic reference · source ID wp-complex-number
nLab — complex number — complex number · mathematical reference · source ID nlab-complex-number

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.