Do you have a easy proof of this affirmation: $\dim_K(k[x_1,...,x_n])=n$ please ? Cause I found some proofs but they all use others theorems I don't know... Thank you !
P.S: Sorry if I made mistakes, I am not English but French..
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You might want to give your own ideas, because if you don't, we won't know which proofs you know and which ones you don't... Obviously, it suffices to show $\dim(R[x]) = \dim(R) + 1$ for every ring (maybe domain or some other nice properties) $R$. – Dirk Jul 03 '17 at 12:36
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Yes of course this theorem is perfect, but do you know how to prove it ? It is when R in a Noetherian ring. – bobito Jul 03 '17 at 12:44
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There are multiple ways to prove it and you claim that you already know some of them. So as long as you don't state clearly which theorems you know and can use and which you don't, I will most likely also just use things you don't know... – Dirk Jul 03 '17 at 12:52
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1This is Proposition 11.9 of these notes by Andreas Gathmann. The chapter is mostly self-contained, but makes some references to other chapters of his notes, which can be found here. – Viktor Vaughn Jul 03 '17 at 20:07
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The result is certainly non-trivial and any proof is going to depend on other results. – Mariano Suárez-Álvarez Jul 03 '17 at 20:10
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I think it's a duplicate of one of the most famous MSE questions. – Pierre-Yves Gaillard Jul 03 '17 at 20:52
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Sorry guys I just saw your answers! I'll have a look at it!! Thanks a lot :) – bobito Jul 16 '17 at 16:33
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@André3000, please could you edit your comment? I am interested, but the link is not available. Many thanks in advance! – Quiet_waters Dec 06 '19 at 16:55
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1@Na'omi I think these links work: https://www.mathematik.uni-kl.de/~gathmann/class/commalg-2013/commalg-2013.pdf https://www.mathematik.uni-kl.de/~gathmann/en/commalg.php – Viktor Vaughn Dec 06 '19 at 18:42
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@André3000 thank you! – Quiet_waters Dec 06 '19 at 19:30