$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$

Depends on

• The complex numbers as $\mathbb R[x]/(x^2+1)$, with the real embedding and imaginary unit $i$
• $x^2+1$ is irreducible over $\mathbb R$
• $F[x]/(p)$ for monic irreducible $p$ is a field extension containing the root $x+(p)$ with unique reduced representatives
• The reals form a totally ordered field
• Squares of nonzero elements are positive

Used by

• ℂ/ℝ has power basis 1,i and degree 2 Corollary
• Complex de Moivre formula for every integer exponent Corollary
• For n≥2, the sum of all n-th roots of unity is zero Corollary
• The complex Jacobian determinant of a composite of equidimensional holomorphic maps is the product Corollary
• The index of a cycle is locally constant off its trace and vanishes far from it Corollary
• The integral of a continuous derivative over a cycle is zero Corollary
• The normalized integral around a positively oriented circle centred at a is 1 Corollary
• The ring of holomorphic functions on a complex domain is an integral domain Corollary
• A nonconstant Blaschke factor has constant boundary modulus Counterexample
• Adjoining one root need not split the polynomial: ℚ(³√2) does not split x³-2 Counterexample
• The disc algebra is unital and separating but not self-adjoint or dense Counterexample
• A complex measure is a finite-valued countably additive set function Definition
• Arithmetic functions on the positive integers Definition
• Character and maximal ideal space Definition
• Complex logarithms, the principal logarithm, and principal and multivalued complex powers Definition
• Complex series, absolute convergence, complex power series, and radius of convergence Definition
• Complex simple functions as finite sums of measurable indicators Definition
• Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential Definition
• Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials Definition
• Holomorphic functions on an open subset of ℂᵐ Definition
• Holomorphic maps ℂᵐ → ℂⁿ and the complex Jacobian matrix Definition
• Integer powers in the complex field Definition
• Integration over a complex chain and the index of a chain Definition
• Multi-indexed power series in ℂᵐ and their absolute convergence Definition
• Real and imaginary parts, complex conjugation, and modulus Definition
• Self-adjoint complex function algebras, unitality, and point separation Definition
• The complex exponential by its power series Definition
• The power functions idₖ and the divisor-power-sum functions σₖ Definition
• Wirtinger operators in ℂᵐ Definition
• ℂ⊗_ℝℂ≅ℂ×ℂ as ℝ-algebras Example
• The complex geometric power series has radius 1 and sums to 1/(1-z) for |z|<1 Example
• The geometric series has only one singular point on its unit circle Example
• The power series of z₀/(1-z₁) and the shape of its domain of convergence Example
• The real 2-dimensional irreducible representation of C₃ has endomorphism ring ℂ Example
• The relative algebraic closure of ℝ in ℂ is all of ℂ Example
• The splitting field of x³-2 over ℚ is ℚ(³√2,ω) with ω=(-1+i√3)/2 Example
• The splitting field of x⁴+2x²-8 over ℚ is ℚ(√2,i) Example
• The square roots of i are ±(1+i)/√2 Example
• The standard inner products make K n, ell two and quotient L two Hilbert spaces Example
• Trigonometric polynomials are uniformly dense on the unit circle Example

…and 24 more results.

Cited to discharge well-definedness by The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i.

Dependency tree · two levels

20 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)