Can we construct the complex numbers by using $i^2 = -1$ as an axiom? Informally we can construct the complex numbers by starting with the real numbers and adding a new element $i$ which satisfy $i^2 = -1$. If we want to formalize this construction we usually use quotient fields, which is analogous to the informal construction, but a bit different on a technical level. However the informal construction still works and is enough for us to start doing computations and algebra with complex numbers. I was wondering if it was possible to make it formal, e.g like this:
We define an algebraic structure $ℂ$ such that:

*

*$ℂ$ is a field.


*$ℝ$ is a subfield of $ℂ$.


*There exist an element $i$ in $ℂ$ such that $i*i=-1$
Does this work as a rigorous construction? To me this doesn't look any less rigorous then e.g the peano axioms, and it captures everything from the informal construction.

Define the complex numbers as $\Bbb C =\{a+bi\mid a,b\in\Bbb R\}$, where the sums are just formal sums (without evaluation).
Define addition componentwise as
$$(a+bi)+(c+di) = (a+c)+(b+d)i$$
and multiplication (multiply out)
$$(a+bi)\cdot(c+di) = ac+adi+bci+bdi^2$$
which becomes with $i^2=-1$
$$(ac-bd)+(ad+bc)i$$
One can show that $\Bbb C$ is a field.
