affine space

Definition.

Let $K$ be a field and let $n$ be a positive integer. In algebraic geometry we define affine space (or affine $n$-space) to be the set

$$\{(k_1,\mathrm{\dots },k_n):k_i\in K\}.$$

Affine space is usually denoted by $K^n$ or ${\mathrm{A}}^n$ (or ${\mathrm{A}}^n\mathit{}\mathrm{(}K\mathrm{)}$ if we want to emphasize the field of definition).

In Algebraic Geometry, we consider affine space as a topological space, with the usual Zariski topology (see also algebraic set, affine variety). The polynomials in the ring $K[x_1,\mathrm{\dots },x_n]$ are regarded as functions (algebraic functions) on ${𝔸}^n(K)$. “Gluing” several copies of affine space one obtains a projective space.

Lemma.

If $K$ is algebraically closed, affine space ${\mathrm{A}}^n\mathit{}\mathrm{(}K\mathrm{)}$ is an irreducible algebraic variety.