On the definition of algebraic dimension

Question

In the book of complex geometry by D. Huybrechts, the algebraic dimension of a compact connected complex manifold $X$ is $\mathrm{a}(X):=\mathrm{trdeg}\mathcal{K}(X)$ over $\mathbb{C}$, where $\mathbb{C}$ is the complex number field and $\mathcal{K}(X)$ the function field of $X$. By Siegel's theorem, one knows that $\mathrm{a}(X)\leq \mathrm{dim}(X)$, so what if we omit the condition of compactness? In the case, how to define algebraic dimension of a connected complex manifold? In other words, my question is that whether the compactness condition of above definition is necessary or not. Thanks in advance!