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

Depends on

• The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
• The quotient ring $R/I$ with $(r+I)(s+I)=rs+I$
• Field extensions, generated subrings $F[S]$, generated subfields $F(S)$, and simple extensions

Used by

• Flexible GMRES with variable right preconditioning Algorithm
• A square root of -1 in a real field extension determines a unique real-field homomorphism from ℂ Corollary
• ℂ/ℝ has power basis 1,i and degree 2 Corollary
• Realification doubles finite dimension Corollary
• A continuous function of modulus at most one need not be positive definite Counterexample
• A first resolvent estimate does not ensure the prescribed semigroup bound Counterexample
• The integers do not have property (T) Counterexample
• Analytic topology and finite smooth algebraic stratifications of complex varieties Definition
• Complex norm and complex normed space Definition
• Complexification as ℂ⊗_ℝV with its canonical real-linear embedding Definition
• Linear multistep methods, consistency, order, and the root condition Definition
• Mackey Borel structure and countable separation of the unitary dual Definition
• Polygonal length and rectifiability of complex paths Definition
• Positive definite functions on an abelian group Definition
• Rational mixed Hodge structures Definition
• Real, complex, and quaternion coefficient algebras and matrix notation Definition
• Realification of a complex vector space by restriction of scalars Definition
• Scalar stability function, A-stability, and L-stability for Runge–Kutta methods Definition
• Self-adjoint complex function algebras, unitality, and point separation Definition
• The complex exponential by its power series Definition
• Uniform periodic, infinite, and Dirichlet grid conventions Definition
• {1+i, 1-i} is a normal basis of ℂ/ℝ while {1,i} is not Example
• A finite coordinate path and a bounded nonrectifiable path Example
• A finite-matrix Schrödinger split preserves norm Example
• A mixed Hodge extension without a rational splitting Example
• A real norm on C that is not a complex norm Example
• ℂ⊗_ℝℂ≅ℂ×ℂ as ℝ-algebras Example
• Equispaced interpolation of Runge's function diverges Example
• GNS representation of a continuous unitary character Example
• Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
• Quarter-turn rotation is not diagonalisable over ℝ but is diagonalisable over ℂ Example
• Simple unit-circle roots give bounded oscillatory modes Example
• Square and hexagonal lattice invariants Example
• The A₂ discriminant, its Jacobian and the top coinvariant class in ℂ[u,z]/(uz,u³+z³) Example
• The exceptional spectra for E₆ and H₃ computed exactly: characteristic polynomials, cyclotomic factorisations and the resulting degree tables Example
• The real 2-dimensional irreducible representation of C₃ has endomorphism ring ℂ Example
• The two-point unordered cover of the plane and the monodromy of a half turn Example
• Two ordered points in the plane: centre and difference coordinates Example
• FALSE: the complex field admits an order making it an ordered field False statement
• Canonical Banach complexification of a real Banach space Lemma

…and 17 more results.

Dependency tree · two levels

13 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

• T. Judson, Abstract Algebra: Theory and Applications, Extension Fields (standard reference, not scraped)