Complex Numbers as Quotient Ring of Real Polynomial
Theorem
Let $\C$ be the set of complex numbers.
Let $P \left[{x}\right]$ be the set of polynomials over real numbers, where the coefficients of the polynomials are real.
Let $\left\langle{x^2 + 1}\right\rangle = \left\{ {Q \left({x}\right) \left({x^2 + 1}\right): Q \left({x}\right) \in P \left[{x}\right]}\right\}$ be the ideal generated by $x^2 + 1$ in $P \left[{x}\right]$.
Let $D = P \left[{x}\right] / \left\langle{x^2 + 1}\right\rangle$ be the quotient of $P \left[{x}\right]$ modulo $\left\langle{x^2 + 1}\right\rangle$.
Then:
- $\left({\C, +, \times}\right) \cong \left({D, +, \times}\right)$
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:
- $\phi \left({\left[\!\left[{a + b x}\right]\!\right]}\right) = a + b i$
We have that:
- $\forall z = a + b i \in \C : \exists \left[\!\left[{a + b x}\right]\!\right] \in D$
such that:
- $\phi \left({\left[\!\left[{a + b x}\right]\!\right]}\right) = a + b i = z$
So $\phi$ is a surjection.
To prove that it is a injection, we let:
- $\phi \left({\left[\!\left[{a + b x}\right]\!\right]}\right) = \phi \left({\left[\!\left[{c + d x}\right]\!\right]}\right)$
So:
| \(\displaystyle \phi \left({\left[\!\left[{a + b x}\right]\!\right]}\right)\) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{c + d x}\right]\!\right]}\right)\) | |||||||||||
| \(\displaystyle \iff \ \ \) | \(\displaystyle a + b i\) | \(=\) | \(\displaystyle c + d i\) | Definition of $\phi$ | |||||||||
| \(\displaystyle \iff \ \ \) | \(\displaystyle a = c\) | \(\land\) | \(\displaystyle b = d\) | Equating both real part and imaginary part | |||||||||
| \(\displaystyle \iff \ \ \) | \(\displaystyle a + b x\) | \(=\) | \(\displaystyle c + d x\) | ||||||||||
| \(\displaystyle \iff \ \ \) | \(\displaystyle \left[\!\left[{a + b x}\right]\!\right]\) | \(=\) | \(\displaystyle \left[\!\left[{c + d x}\right]\!\right]\) |
So $\phi$ is an injection and thus a bijection.
It remains to show that $\phi$ is a homomorphism for the operation $+$ and $\times$.
| \(\displaystyle \phi \left({\left[\!\left[{a + b x}\right]\!\right] + \left[\!\left[{c + d x}\right]\!\right]}\right)\) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{\left({a + c}\right) + \left({b + d}\right) x}\right]\!\right]}\right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a + c}\right) + \left({b + d}\right) i\) | Definition of $\phi$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a + b i}\right) + \left({c + d i}\right)\) | Definition of Complex Addition | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{a + b x}\right]\!\right]}\right) + \phi \left({\left[\!\left[{c + d x}\right]\!\right]}\right)\) |
| \(\displaystyle \phi \left({\left[\!\left[{a + b x}\right]\!\right] \times \left[\!\left[{c + d x}\right]\!\right]}\right)\) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{\left({a + b x}\right) \times \left({c + d x}\right)}\right]\!\right]}\right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{a \times c + \left({a \times d + b \times c}\right) x + b \times d \, x^2}\right]\!\right]}\right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{a \times c + \left({a \times d + b \times c}\right) x + b \times d \, x^2 - b \times d \left({x^2 + 1}\right)}\right]\!\right]}\right)\) | Definition of $D$ as a quotient ring modulo $\left\langle{x^2 + 1}\right\rangle$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{\left({a \times c - b \times d}\right) + \left({a \times d + b \times c}\right) x}\right]\!\right]}\right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a \times c - b \times d}\right) + \left({a \times d + b \times c}\right) i\) | Definition of $\phi$ | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({a + b i}\right) \times \left({c + d i}\right)\) | Definition of Complex Multiplication | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({\left[\!\left[{a + b x}\right]\!\right]}\right) \times \phi \left({\left[\!\left[{c + d x}\right]\!\right]}\right)\) | Definition of $\phi$ |
Thus $\phi$ has been demonstrated to be a bijective ring homomorphism and thus by definition a ring isomorphism.
$\blacksquare$
