How to describe Algebraic Closure of $\mathbb{C}(x)$?

Let $\mathbb{C}$ be the set of complex numbers, and $x$ be an indeterminate. Let $\overline{\mathbb{C}(x)}$ be an algebraic closure of $\mathbb{C}(x)$. Then what are the elements of $\overline{\mathbb{C}(x)}$?

Obviously, elements like $\sqrt[n]{x}, \sqrt[n]{f(x)}$ (where $f(x)\in\mathbb{C}(x)$), etc. will be in the set. But is it easy to describe the full set?

Solution 1:

Yes, it possible to describe the elements of $\overline{\mathbb{C}(x)}$ very explicitly.
They are called Puiseux series and consist of infinite series of the form $$ax^{-1/12}+bx^{-7/3}+cx^{4/11}+\ldots$$ with $a,b, c,\ldots \in \mathbb{C}$. The infinitely many fractions which appear as exponents must have a common denominator and only finitely many of them may be negative.The rules for addition and multiplication are the evident ones, analogous to those for ordinary power series with integral exponents.
Of course this is an informal definition: officially you decree that a Puiseux series is a map $\mathbb Q \to \mathbb C$ whose support (=set of rationals where the map is $\neq 0$) is both bounded below and in $\frac{1}{n} \mathbb Z$ for some integer $n\gt 0$ (depending on the series).
The key result is that the Puiseux series form an algebraically closed field.
Hence the set of Puiseux series algebraic over $\mathbb{C}(x)$ gives an algebraic closure $\overline{\mathbb{C}(x)}$ of $\mathbb C(x)$.
For example, the equations $y^3=x$ or $z^2=1+x$ have solutions $y=x^{1/3}$ and $z=\Sigma_{n\geq 0}\binom {1/2}{n}x^n$.
The exact same procedure describes an algebraic closure of $F(x)$ for $F$ an arbitrary algebraically closed field of characteristic zero.

Here is a link to the Wikipedia article on the subject.

Solution 2:

It really depends on what counts as a description. e.g. for some purposes, "the algebraic closure of $\mathbb{C}(x)$" is already a rather good and easy description!

For explicit computation, it's often good enough to just construct extension fields of $\mathbb{C}(x)$ as you need them -- e.g. to start, you might decree that $\alpha$ satisfies $f(x, \alpha) = 0$, and then continue by working in the field $\mathbb{C}(x, \alpha)$ (which is, of course, isomorphic to $\mathbb{C}(x)[y] / f(x, y)$.

As a twist on this, there's probably some way to use Riemann surfaces (together with a local choice of branch) embedded in $\mathbb{C}^2$ to name elements of the algebraic closure, but it's not immediately obvious to me if it would work.

The field of complex Puiseaux series is not an algebraic closure -- it is the algebraic closure of $\mathbb{C}((x))$ and thus "too big" -- but it does contain an algebraic closure of $\mathbb{C}(x)$, so you can name things this way.

As an aside, writing things like $\sqrt[n]{x}$ is tricky, since the algebraic closure actually contains $n$ elements that can rightfully claim that name, and without choosing a specific representation of the set, it is impossible to specify which of those $n$ elements you mean. (Of course, you can get around this by simply decreeing that we choose one before-hand, and use $\sqrt[n]{x}$ to refer to it)