Complex Numbers as Quotient Ring of Real Polynomial
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.
$\blacksquare$