# A question on the construction of the complex numbers and field extensions

I am thinking about a construction of the complex numbers. I know that it is not necessary: for a complex analysis course one could just give the field axioms and then take $\mathbb C$ to be a field that satisfies them.

But since we can construct the real numbers as set of equivalence classes of Cauchy sequences in $\mathbb Q$ I started to think about how to do a construction for $\mathbb C$.

I found that $\mathbb C$ can be constructed as the field extension $\mathbb R[x]/\langle x^2 + 1\rangle$ of $\mathbb R$.

My question is:

Does this construction give a unique extension field? (are extension fields unique? I could not find any information in the affirmative)

And if not, how can one prove after constructing $\mathbb C$ like this that $\mathbb C$ is the unique field with the stipulated properties?

• Up to isomorphism, $\mathbb C$ is the only finite extension of $\mathbb R$. This follows from the fact that $\mathbb C$ is algebraically closed, that is, the fundamental theorem of algebra.
• There are many others extensions of $\mathbb R$, such as $\mathbb R(X)$, the field of rational functions with real coefficients.
Since $\Bbb C$ is an algebraic closure of $\Bbb R$, and algebraic closures are unique up to isomorphism over the base field, that makes $\Bbb C$ the unique algebraic extension up to $\Bbb R$-isomorphism. That is, if we have two algebraic closures $K_1$ and $K_2$ of a field $k$, there exists a $K_1\xrightarrow{\sim} K_2$ which restricts to $k\to k$.
One learns there are two basic types of extensions: algebraic and transcendental. The first occurs when one adjoins elements that satisfy algebraic relations, the second occurs when one adjoins transcendentals. The field $\Bbb R(X)$ of rational functions in an indeterminate $X$ has the element $X$ which satisfies no algebraic relation over $\Bbb R$ (since distinct rational functions are distinct elements of $\Bbb R(X)$), so this is such an example. Any extension can be split into an algebraic extension of a purely transcendental extension.
In general, extensions are not unique. The rationals $\Bbb Q$ have infinitely many algebraic extensions that lie between it and its algebraic closure $\overline{\Bbb Q}$. Some of them are isomorphic but distinct even, for instance $\Bbb Q(\sqrt{2})\cong\Bbb Q[X]/(X^3-2)\cong \Bbb Q(e^{2\pi i/3}\sqrt{2})$ (the isomorphisms preserve $\Bbb Q$) but these two fields $\Bbb Q(\sqrt{2})\ne\Bbb Q(e^{2\pi i/3}\sqrt{2})$ are not equal since one has complex numbers and the other doesn't.