Maximal Ideals of $F[x_{1} , \dots , x_{n}]$, $F$ not necessarily algebraically closed 
Suppose that $F$ is a field (not necessarily algebraically closed) and $R = F[x_{1} , \dots , x_{n}]$ is the ring of polynomials in $n$-variables. How can I classify the maximal ideals of $R$? 

If $F = \overline{F}$, then the weak nullstellensatz, states that the maximal ideals of $R$ are the sets $I = (x_{1} - a_{1} , \dots , x_{n} - a_{n}) : a_{i} \in F , \forall i \in [n]$. 

Is it the case that the maximal ideals of $R$ when $F$ is not algebraically closed are $I = \{ f \in R : f(a_{1} , \dots , a_{n}) = 0 \} \text{ for some } a_{i} \in \overline{F}$? If so how does one prove it? 

Could I argue that $I \subset \overline{I}$, where $\overline{I} \unlhd \overline{F}[x_{1} , \dots , x_{n}]$ is a maximal ideal of the ring of polynomials in $n$ -variables over the algebraic closure of $F$? However, I believe this argument does not consider the fact that $I$ is maximal. 
 A: The characterization is correct. Let $M$ be a maximal ideal of $F[x_1,\cdots,x_n]$. Then $F[x_1,\cdots,x_n]/M=F'$ is a field, and it's finitely generated as an algebra over $F$. Therefore $F'$ is a field extension of finite degree, and thus algebraic. Through embedding $F'\hookrightarrow \overline{F}$, extending the natural embedding $F\hookrightarrow \overline{F}$ to an embedding $F[x_1,\cdots,x_n]\hookrightarrow \overline{F}[x_1,\cdots,x_n]$, and defining a map $\overline{F}[x_1,\cdots,x_n]\to\overline{F}$ which takes $x_i\in \overline{F}[x_1,\cdots,x_n]$ to the image of $x_i$ under the composite map $F[x_1,\cdots,x_n]\to F'\to \overline{F}$, we can consider the following commutative diagram of rings:
$$\require{AMScd}
\begin{CD}
F[x_1,\cdots,x_n] @>{}>> F'\\
@VVV @VVV \\
\overline{F}[x_1,\cdots,x_n] @>{}>> \overline{F}
\end{CD}$$
Clearly the bottom map corresponds to a maximal ideal of $\overline{F}[x_1,\cdots,x_n]$, which must be of the form $(x_1-a_1,\cdots)$ for $a_i\in \overline{F}$ by the Nullstellensatz - equivalently, it is all polynomials in $\overline{F}[x_1,\cdots,x_n]$ which vanish on $(a_1,\cdots,a_n)\in\overline{F}^n$. The preimage of this maximal ideal inside $F[x_1,\cdots,x_n]$ is also maximal, and may be described exactly as the polynomials in $F[x_1,\cdots,x_n]$ vanishing on $(a_1,\cdots,a_n)\in\overline{F}^n$. So we have shown the requested characterization.
