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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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  • $\begingroup$ Not in general, consider $\mathbb{A}^1(\mathbb{F}_3)=\mathbb{F}_3$. It is the union $V(x) \cup V((x-1)(x-2))$ $\endgroup$ – rVitale Dec 8 '15 at 21:10
  • $\begingroup$ Thank you. :-) What about $ \mathbb{P}^n (k) $ ? What happens when $ k = \mathbb{C} $ ? $\endgroup$ – Lina45 Dec 8 '15 at 21:22
  • $\begingroup$ When $k=\mathbb{C}$ or any infinite field those spaces are irreducible for $n \geq 1$ There's a nice discussion here $\endgroup$ – rVitale Dec 8 '15 at 21:31
  • $\begingroup$ Thank you very much rVitale. :-) $\endgroup$ – Lina45 Dec 8 '15 at 21:36
  • $\begingroup$ No problem glad it helped! $\endgroup$ – rVitale Dec 8 '15 at 21:43

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