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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?

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  • 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.

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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 $k\to k$.

It is certainly not the only extension - we have ways of constructing extensions arbitrarily, indeed there are extensions of any field of every possible cardinality, so in particular the isomorphism classes of extensions is too big to even be a set!

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 $\Bbb Q(\sqrt[3]{2})\cong\Bbb Q[X]/(X^3-2)\cong \Bbb Q(e^{2\pi i/3}\sqrt[3]{2})$ (the isomorphisms preserve $\Bbb Q$) but these two fields $\Bbb Q(\sqrt[3]{2})\ne\Bbb Q(e^{2\pi i/3}\sqrt[3]{2})$ are not equal since one has complex numbers and the other doesn't.

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