What does $\Bbb{C}(X)$ refer to?

Question

I have from a book

(b) Let $E = \Bbb{C}(X)$. Then $\operatorname{Aut}(E / \Bbb{C})$ consists of the maps $X \mapsto \dfrac{aX + b}{cX + d}, ad-bc \neq 0\ldots$

Not sure what $\Bbb{C}(X)$ is. Thanks.

Answer 1

Well, first, what is $\mathbb{C}[X]$? It is, by definition, the set $\{ a_{0} + a_{1}X + a_{2} X^{2} + a_{3}X^{3} + \dots + a_{n}X^{n} | a_{i} \in \mathbb{C}, n \in \Bbb N \}$. In other words, it is the ring of polynomials with coefficients in $\Bbb C$ (and it's actually an integral domain).

Now, $\Bbb C(X)$ is the fractional field of the integral domain $\mathbb{C}[X]$. That is, $\Bbb C(X)$ is the set of elements that look like $\frac{a}{b}$ where $a, b \in \mathbb{C}[X]$ and $b \neq 0$. Of course, we are really talking about equivalence classes (just like in $\mathbb{Q}$, $\frac{2}{3} = \frac{4}{6}$, for example). It's easy to prove that $\mathbb{C}(X)$ satisfies the axioms of a field. We call it the fractional field of the ring $\Bbb C[X]$ since it's constructed by taking all possible "fractions" made from the elements of $\Bbb C[X]$.

Answer 2

$E = \mathbf{C}(X)$ is the fractions field of the domain $\mathbf{C}[X]$, ring of polynomials in the formal variable $X$ with coefficients in $\mathbf{C}$. Every element of $E$ can be written $\frac{P}{Q}$ with $P,Q\in\mathbf{C}[X]$ coprime polynomials and $Q\not=0$, which in this case means that $P$ and $Q$ have no common roots. The definition is valid for any field $k$ instead of $\mathbf{C}$.

Answer 3

It is the field of rational functions with complex coefficients in one variable. It is the quotient field of the ring of polynomials in one variable, that is an element of it is the quotient of two polynomials.

Answer 4

$\mathbb{C}(X)$ is the field of rational functions in the indeterminate $X$ over $\mathbb{C}$. This is the field consisting of all quotients of polynomials in $X$ (with the denominator nonzero) with coefficients in $\mathbb{C}$.