Complex Numbers as Quotient Ring of Real Polynomial

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Theorem

Let $\C$ be the set of complex numbers.

Let $P \sqbrk x$ be the set of polynomials over real numbers, where the coefficients of the polynomials are real.

Let $\ideal {x^2 + 1} = \set {\map Q x \paren {x^2 + 1}: \map Q x \in P \sqbrk x}$ be the ideal generated by $x^2 + 1$ in $P \sqbrk x$.

Let $D = P \sqbrk x / \ideal {x^2 + 1}$ be the quotient of $P \sqbrk x$ modulo $\ideal {x^2 + 1}$.


Then:

$\struct {\C, +, \times} \cong \struct {D, +, \times}$


Proof

By Division Algorithm of Polynomial, any set in $D$ has an element in the form $a + b x$.

Define $\phi: D \to \C$ as a mapping:

$\map \phi {\eqclass {a + b x} {x^2 + 1} } = a + b i$

We have that:

$\forall z = a + b i \in \C : \exists \eqclass {a + b x} {x^2 + 1} \in D$

such that:

$\map \phi {\eqclass {a + b x} {x^2 + 1} } = a + b i = z$

So $\phi$ is a surjection.


To prove that it is a injection, we let:

$\map \phi {\eqclass {a + b x} {x^2 + 1} } = \map \phi {\eqclass {c + d x} {x^2 + 1} }$

So:

\(\ds \map \phi {\eqclass {a + b x} {x^2 + 1} }\) \(=\) \(\ds \map \phi {\eqclass {c + d x} {x^2 + 1} }\)
\(\ds \leadstoandfrom \ \ \) \(\ds a + b i\) \(=\) \(\ds c + d i\) Definition of $\phi$
\(\ds \leadstoandfrom \ \ \) \(\ds a = c\) \(\land\) \(\ds b = d\) Equality of Complex Numbers
\(\ds \leadstoandfrom \ \ \) \(\ds a + b x\) \(=\) \(\ds c + d x\)
\(\ds \leadstoandfrom \ \ \) \(\ds \eqclass {a + b x} {x^2 + 1}\) \(=\) \(\ds \eqclass {c + d x} {x^2 + 1}\)

So $\phi$ is an injection and thus a bijection.


It remains to show that $\phi$ is a homomorphism for the operation $+$ and $\times$.

\(\ds \map \phi {\eqclass {a + b x} {x^2 + 1} + \eqclass {c + d x} {x^2 + 1} }\) \(=\) \(\ds \map \phi {\eqclass {\paren {a + c} + \paren {b + d} x} {x^2 + 1} }\)
\(\ds \) \(=\) \(\ds \paren {a + c} + \paren {b + d} i\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \paren {a + b i} + \paren {c + d i}\) Definition of Complex Addition
\(\ds \) \(=\) \(\ds \map \phi {\eqclass {a + b x} {x^2 + 1} } + \map \phi {\eqclass {c + d x} {x^2 + 1} }\)


\(\ds \map \phi {\eqclass {a + b x} {x^2 + 1} \times \eqclass {c + d x} {x^2 + 1} }\) \(=\) \(\ds \map \phi {\eqclass {\paren {a + b x} \times \paren {c + d x} } {x^2 + 1} }\)
\(\ds \) \(=\) \(\ds \map \phi {\eqclass {a \times c + \paren {a \times d + b \times c} x + b \times d \, x^2} {x^2 + 1} }\)
\(\ds \) \(=\) \(\ds \map \phi {\eqclass {a \times c + \paren {a \times d + b \times c} x + b \times d \, x^2 - b \times d \paren {x^2 + 1} } {x^2 + 1} }\) Definition of $D$ as a quotient ring modulo $\ideal {x^2 + 1}$
\(\ds \) \(=\) \(\ds \map \phi {\eqclass {\paren {a \times c - b \times d} + \paren {a \times d + b \times c} x} {x^2 + 1} }\)
\(\ds \) \(=\) \(\ds \paren {a \times c - b \times d} + \paren {a \times d + b \times c} i\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \paren {a + b i} \times \paren {c + d i}\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds \map \phi {\eqclass {a + b x} {x^2 + 1} } \times \map \phi {\eqclass {c + d x} {x^2 + 1} }\) Definition of $\phi$


Thus $\phi$ has been demonstrated to be a bijective ring homomorphism and thus by definition a ring isomorphism.

$\blacksquare$