Prove that these two fields are isomorphic.

Question

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a variety, so it is prime.

Now, on the other hand:

$$M_p=\{f \in \bar{K}[V] : f(p)=0\}.$$

So we have to find a bijective function from this two fields (may be there are better ways to do this, but I am not quite sure if in field theory we can find that machinery), but my intuition says that in $\bar{K}[V]/M_p$ we don't have any vanishing polynomial at $V$ so I can't figure out how to find this bijection.

Can someone help me with this problem?

Thanks a lot in advance.

I don't understand your question. Firstly $\tilde{K}[x_1,\dotsc,x_n]$ is not a field. Secondly you define $\tilde{K}[V]$ as a quotient of $\tilde{K}[x_1,\dotsc,x_n]$, and then take its quotient by another ideal of $\tilde{K}[x_1,\dotsc,x_n]$ which doesn't make sense. — Josh, Feb 12 '16 at 02:36
I think you may have some typos in your question then :) See my first comment. — Josh, Feb 12 '16 at 02:38
I know it is a little bit weird to have those quotients but I've got it like that :( — user162343, Feb 12 '16 at 02:44

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

1

If $P=(a_1,\dots,a_n)$, then $M_P=(x_1-a_1,\dots,x_n-a_n)/I(V)$; see Proving that kernels of evaluation maps are generated by the $x_i - a_i$ (the proof given there works for any field).

Then $\overline K[V]/M_P\simeq\overline K[x_1,\dots,x_n]/(x_1-a_1,\dots,x_n-a_n)\simeq\overline K$; for the last isomorphism see Maximal ideals in $K[X_1,\dots,X_n]$.

edited Apr 13 '17 at 12:21

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answered Feb 12 '16 at 16:43

user26857

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Prove that these two fields are isomorphic.

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