When I am reading Atiyah-Macdonald's Introduction to Commutative Algebra, the author wants to establish the equality of the Krull dimension of the localized coordinate ring at a point and the dimension of the variety $V$. That is, for irreducible affine variety $V$, let the coordinate ring be $A(V)=k[x_1,...,x_n]/I(V)$, then the author wants to prove that $$\text{trdeg}(\text{Frac}(A(V)/k)) = \dim (A(V)_m)$$ for every maximal ideal $m$ of $A(V).$ I have a problem with the $\geq$ part. To prove the $\geq$ direction, he uses the following lemma:
If $k\subseteq A$ field mapping isomorphically onto $A/m$ and if $x_1,...,x_d$ is a system of parameters ($d = \dim A $ and $(x_1,...,x_d)$ generates a $m-$primary ideal), then $x_1,...,x_d$ are algebraically independent over $k.$
I can understand that the author wants to use this lemma to create a system of parameters $x_1,...,x_d \in A(V)_m$ that are algebraically independent over $k$. Since $d = \dim (A(V)_m)$, we can conclude $\text{trdeg}(\text{Frac}(A(V)/k)) \geq \dim (A(V)_m)$. The problem is, we then have to prove that $k\cong A(V)_m/mA(V)_m$ and that $k\subseteq A(V)_m.$ Also, we have to prove the existence of such system of parameters. These do not look trivial to me, can anyone give me some hints or help on them? Any help is very appreciated!
