# Are $\mathbb{A}^n (k ) = k^n$ and $\mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles?

Are $\mathbb{A} (k ) = k^n$ and $\mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when $k$ is a domain ?

• Not in general, consider $\mathbb{A}^1(\mathbb{F}_3)=\mathbb{F}_3$. It is the union $V(x) \cup V((x-1)(x-2))$ – rVitale Dec 8 '15 at 21:10
• Thank you. :-) What about $\mathbb{P}^n (k)$ ? What happens when $k = \mathbb{C}$ ? – Lina45 Dec 8 '15 at 21:22
• When $k=\mathbb{C}$ or any infinite field those spaces are irreducible for $n \geq 1$ There's a nice discussion here – rVitale Dec 8 '15 at 21:31