I am studying Stillwell's Elements of Algebra. In Chapter 3, Section 3.6, he writes
There is a unique extension of $+$, $-$, $\times$, $\div$ to $\mathbb C$ satisfying the field properties since if these properties hold we necessarily have \begin{align*} (\alpha_1 + i\beta_1) + (\alpha_2+ i\beta_2) & = (\alpha_1 + \alpha_2) + i(\beta_1 + \beta_2)\text{,}\\ (\alpha_1 + i\beta_1)(\alpha_2 + i\beta2) & = (\alpha_1\alpha_2 - \beta_1\beta_2) + i(\alpha_1\beta_2 + \beta_1\alpha_2) \end{align*} (using the fact that $i^2 = -1$).
Now what I understood from this is the following.
If we try to make $\mathbb R\times\mathbb R$ a field such that for each $\alpha_1, \alpha_2\in\mathbb R$, we have
- "extension" of addition and multiplication from $\mathbb R$ so that
- $(\alpha_1, 0) + (\alpha_2, 0) = (\alpha_1 + \alpha_2, 0)$, and
- $(\alpha_1, 0)(\alpha_2, 0) = (\alpha_1\alpha_2, 0)$,
- "$i^2 = -1$", that is,
- $(0, 1)(0, 1) = (-1, 0)$,
then we must have the familiar definitions of the addition and product rules for $\mathbb C$. (Note that I have denoted the addition and multiplication in $\mathbb R$ and $\mathbb{R\times R}$ by same notations.)
I have realized that I just need to show that for any $(\alpha, \beta)\in\mathbb{R\times R}$, we have $(\alpha, \beta) = (\alpha, 0) + (0, 1)(\beta, 0)$.
Question: Can someone please help showing this? Thanks for your time!