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 ?
Thanks in advance for your help.
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Not in general, consider $\mathbb{A}^1(\mathbb{F}_3)=\mathbb{F}_3$. It is the union $V(x) \cup V((x-1)(x-2))$ – Ryan Vitale Dec 08 '15 at 21:10
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Thank you. :-) What about $ \mathbb{P}^n (k) $ ? What happens when $ k = \mathbb{C} $ ? – Lina45 Dec 08 '15 at 21:22
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When $k=\mathbb{C}$ or any infinite field those spaces are irreducible for $n \geq 1$ There's a nice discussion here – Ryan Vitale Dec 08 '15 at 21:31
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Thank you very much rVitale. :-) – Lina45 Dec 08 '15 at 21:36
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No problem glad it helped! – Ryan Vitale Dec 08 '15 at 21:43