Possible dimension of residue fields in $k[x_1,x_2,\ldots]$ My motivation comes from Nullstellensatz, a particular form of which is the following:
$(\ast)$  All residue fields of $k[x_1,\ldots,x_n]$ must be a finite extension of $k$.
So the natural extension is:
What dimension (over $k$) of residue fields are possible for infinitely many variables i.e  $k[x_1,x_2,\ldots] $ or more generally in $k[x_i\mid i\in I]$.
Remarks:
For the time being let's focus on countably many indeterminates.
(1) For $k$ finite, it is possible to find residue fields of dimension $1\leq n\leq |\mathbb{N}|$  (and perhaps only these dimensions are possible but I am not sure).
(2) For $k$ of countable size, it is possible to find residue fields of dimension $|\mathbb{N}|$ and some finite dimensions are also possible depending on $k$. For example if $k$ is algebraically closed then no finite dimension is possible but if $k=\mathbb{R}$ residue field of dimension $2$ is possible.
(3) For $k$ of uncountable size, finite dimensions will depend on $k$ and I am not sure of whether dimension $|\mathbb{N}|$ is possible or not .
If $K$ (the residue field) is not algebraic over $k$, by $\dim_kK\geq |k|$, it would be forced to be of uncountable dimension but then again there are many such sizes and I don't know if all are possible.   
 A: It is impossible to have dimension greater than $ |\mathbb N| = \aleph_0 $, since $ k[x_1, x_2, \ldots] $ is itself a countable dimensional vector space over $ k $; and thus any quotient has $ k $-dimension at most $ \aleph_0 $. It's fairly easy to see that $ \bar {\mathbf Q_p} = \mathbf Q_p \bar{\mathbf Q} $ arises as a quotient of $ \mathbf Q_p[x_1, x_2, \ldots] $ (pick an enumeration of the algebraic numbers and map each indeterminate to a distinct algebraic number), and $ \mathbf Q_p $ is uncountable, so dimension $ \aleph_0 $ is possible for uncountable $ k $ as well. In fact, any dimension less than or equal to $ \aleph_0 $ is possible for $ k = \mathbf Q_p $, since $ \mathbf Q_p $ admits extensions of every degree $ n $, for example $ \mathbf Q_p(\sqrt[n]{p})/\mathbf Q_p $ is an extension of degree $ n $. (Map $ x_1 \to \sqrt[n]{p} $ and map everything else to zero for instance...) It is also possible to find countable $ k $ which yield residue fields of every degree $ \leq \aleph_0 $, for example;
$$ k = \bigcup_{q \textrm{ prime}} \mathbf F_{p^{\#q}} $$
will do, where $ \#q $ denotes the primorial.
