Let $k=\mathbb{F}\_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):


1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the (finite) disjoint union of ${\rm Spec}(\kappa(x))$ for all the closed points $x\in X$ of degree $[\kappa(x):k]<N$. For $n< N$, we have a bijection
$${\rm Sym}^ni: ({\rm Sym}^nZ)(k)\xrightarrow{\cong}({\rm Sym}^nX)(k).$$
2. Suppose $X={\rm Spec}(L)$ for a finite extension $L/k$ of degree $d$. Then $({\rm Sym}^nX)(k)$ is empty if $d\nmid n$, and is a singleton if $d\mid n$.


I don't have any idea with them. Can anyone help me to prove these results? Thanks also for any idea/hint/reference!


For the construction of symmetric products of varieties, one can see Section 3.1 in [Hilbert and Chow Schemes of Points,
Symmetric Products and Divided Powers](https://people.kth.se/%7Edary/hilbchowsymdiv-20080416.pdf).



About closed points in symmetric product schemes over a finite field
There are probably many ways to answer your question depending on what your preferred point of view on schemes and symmetric products are. Let me offer the following approach.


Forget that $k$ is a finite field, let it just be a perfect field, call $\newcommand{\alg}{\operatorname{alg}}k^{\alg}$ a fixed chosen algebraic closure and $\Gamma := \newcommand{\Gal}{\operatorname{Gal}}\Gal(k^{\alg}/k)$ its absolute Galois group. We see a $k$-scheme $X$ of finite type through its set $X(k^{\alg})$ of geometric points endowed with a (continuous) action of $\Gamma$; for $k \subseteq k' \subseteq k^{\alg}$, the set $X(k')$ is the set of $\Gamma\_{k'}$-fixed points of $X(k^{\alg})$ where $\Gamma\_{k'} := \Gal(k^{\alg}/k')$. We note that $\newcommand{\Sym}{\operatorname{Sym}}\Sym^n(X)$ has the set $\Sym^n(X(k^{\alg}))$ (of $n$-element multisubsets of $X(k^{\alg})$) as geometric points, with the obvious Galois action. In particular, $(\Sym^n(X))(k)$ is the set of $\Gamma$-stable $n$-element multisubsets of $X(k^{\alg})$. Also note that $\newcommand{\Spec}{\operatorname{Spec}}\Spec(L)$ is seen as the homogeneous $\Gamma$-set $\Gamma/\Gamma\_L$ of left cosets of $\Gamma\_L$ under $\Gamma$.


Now remember that $k$ is a finite field. Then $\Gamma$ is procyclic with progenerator given by the Frobenius $\sigma \colon x \mapsto x^q$ (i.e., a $\Gamma$-set is just a set with a finite-order permutation $\sigma$). When $L$ is the extension of degree $d$, the $\Gamma$-set $\Spec L$ is given by a $d$-cycle; a closed point in $X$ with residue field $L$ is such a $d$-cycle in the Galois action on the geometric points of $X$.


With this in mind, claim (1) essentially says that if $\sigma$ is cyclic permutation on an $n$-element multiset (a $\sigma$-stable $n$-element multisubset of $X(k^{\alg})$), we can see it as a multiset sum of $d$-cycles for $d\leq n$, which we then see as a single $n$-element $\sigma$-stable multisubset of the disjoint union of “archetype” $d$-cycles (one for each $d\leq n$).


As for (2), it says that a $d$-cycle has a single $\sigma$-stable $n$-element multiset when $d|n$ (namely, take every element of the cycle $n/d$ times), and none if $d$ does not divide $n$.


