I am learning introductory algebraic geometry by myself. Probably I am misunderstanding something. Could you point out where I mistake?
Let $k$ be a field that is not necessarily an algebraically closed. Let $V \subset \mathbb{A}^n$ be an algebraic set that is not necessarily irreducible.
We use the notation $\mathcal{I}(V) = \{ f \in k[x_1, \dots, x_n] \mid (\forall p \in V)(f(p) = 0) \}$. We set the coordinate ring of $V$ by $k[V] = k[x_1, \dots, x_n]/\mathcal{I}(V)$. We can regard each element of $k[V]$ as a $k$-valued polynomial function on $V$.
Let $v \in V$. We use another notation $\mathcal{I}(v) = \{ f \in k[V] \mid f(v) = 0 \}$. Since $\mathcal{I}(v)$ is a kernel of a surjective $k$-algebraic homomorphism $\mathrm{ev}_v \colon k[V] \to k$ defined by $\mathrm{ev}_v (f) = f(v)$, we get an isomorphism $k[V] / \mathcal{I}(v) \cong k$. Hence, $\mathcal{I}(v)$ is a maximal ideal of $k[V]$.
We define the local ring of $V$ at $v$ using localization as follows. $$ \mathcal{O}_{v, V} = (k[V] \setminus \mathcal{I}(v))^{-1} k[V]. $$ It is known that $\mathcal{O}_{v, V}$ is a local ring with the unique maximal ideal $$ m_{v, V} = (k[V] \setminus \mathcal{I}(v))^{-1} \mathcal{I}(v). $$ Since $\mathcal{I}(v)$ is a maximal ideal of $k[V]$, we obtain an isomorphism $\mathcal{O}_{v, V} / m_{v, V} \cong k[V] / \mathcal{I}(v) \cong k$.
Dummit-Foote (p.729, Exercise 28) defines that $v \in V$ is a $k$-rational point of $V$ if $\mathcal{O}_{v, V} / m_{v, V} = k$ holds. By the argument above, for any field $k$, for any algebraic set $V$, is every point $v \in V$ $k$-rational point? I am confused now. I would appreciate it if you could help solve my misunderstanding.
P.S. Dummit-Foote defines the local ring of $V$ at $v$ just for the case that $V$ is irreducible and $k$ is algebraically closed. Thus, the definition above is just my guess. If I am wrong, please tell me the correct definition.
P.S.2. I am reading an article on rational point in Wikipedia on English version. I notice that my definition of algebraic set learned from Dummit-Foote is different from Wikipedia. As I wrote, my understanding is that an algebraic set $V$ is a subset of $\mathbb{A}^n = k^n$. But Wikipedia says that $V$ is a subset of $\overline{k}^n$, where $\overline{k}$ denotes an algebraically closed extension of $k$. Therefore, I guess my argument is correct and Dummit-Foote just introduces a technical term "$k$-rational point".