$GL({\mathbb C})$ as an Algebraic Group

Question

An affine algebraic group is an affine variety $G$ such that multiplication and the inverse map are polynomial maps. However, $GL({\mathbb C})$ is given as an example of an algebraic group. Since the formula for inverse of a matrix involves rational functions I do not see how $GL({\mathbb C})$ is well-defined as an algebraic group!

Answer 1

Note that $\mathrm{GL}_n(\mathbb{C})$ does not embed as a Zariski-closed subset into $\mathbb{A}^{n^2}$, but into $\mathbb{A}^{n^2+1}$. A matrix $A \in \mathrm{GL}_n(\mathbb{C})$ is identified with the vector $(A,\det(A)^{-1})$. You can convince yourself, that the inverse map is a polynomial if we regard $\mathrm{GL}_n(\mathbb{C})$ as that subset of $\mathbb{A}^{n^2+1}$.

Also note that affine varieties are not required to be embedded in an affine space. Hence, morphisms between affine varieties may not be polynomials in an obvious way.