All $K$-rational points of $V$ form a $K$-variety

Question

Here is a problem from Ernst Kunz's Introduction to Commutative Algebra and Algebraic Geometry:

Let $L / K$ be an extension of fields, $V \subset \mathbb{A}^n(L)$ an $L$-variety. Then the set $V_k := V \cap \mathbb{A}^n(K)$ of all $K$-rational is a $K$-variety in $\mathbb{A}^n(K)$

Here we can just deal with the case that V is a irreducible hypersurface defined by a irreducible polynomial $F \in L[X_1, ..., X_n]$.

What I've tried:

1. The most luciest case that $F \in K[X_1, ..., X_n]$, then problem solved. If not, suppose F has a zero point in $\mathbb{A}^n(K)$, does there exist a polynomial in $K[X_1, ..., X_n]$ associated with F?($n = 1$ does)

2. Let $\mathfrak{J}_L (V)$ be the set of all polynomials in $L[X_1, ..., X_n]$ which vanish on V, then $\mathfrak{J}_L (V) = (F)$ is an ideal of $L[X_1, ..., X_n]$. Consider $\mathfrak{J}_L (V) \cap K[X_1, ...X_n]$, it may be a zero ideal or a nonzero ideal of $K[X_1, ...X_n]$. For the latter, I want to find the special $h \in K[X_1, ...X_n]$ which $h = FG$ and $h$ does not contribute other zero points.

In fact, I have made no progress in the problem. Anyone can help?

Answer 1

Let $L/K$ be a field extension. Fix a $K$-basis $\{e_\lambda\}_{\lambda\in \Lambda}$ of $L$. Let $V \subseteq \Bbb A^n(L):= L^n$ be an $L$-variety defined by polynomials $F_1,\dotsc, F_r \in L[X_1,\dotsc,X_n]$. We can then write $$ F_i = \sum_{\lambda\in \Lambda} f_{i,\lambda}\cdot e_\lambda, $$ where $f_{i,\lambda}\in K[X_1,\dotsc,X_n]$ for all $i,\lambda$. Notice that $f_{i,\lambda}\neq 0$ only for finitely many pairs $(i,\lambda)$. In other words: there exists a finite subset $J\subseteq \{1,\dotsc,r\}\times\Lambda$ such that $(i,\lambda)\notin J \implies f_{i,\lambda} =0$.

Now, for $x\in \Bbb A^n(K)$ we have the following equivalences: \begin{align*} x\in V\cap \Bbb A^n(K) &\iff F_i(x) = 0 \quad\text{for all $1\le i\le r$}\\ &\iff f_{i,\lambda}(x) = 0 \quad \text{for all $(i,\lambda)\in J$} \end{align*} Therefore, $V\cap \Bbb A^n(K)$ is the $K$-variety defined by the polynomials $f_{i,\lambda} \in K[X_1,\dotsc,X_n]$ for $(i,\lambda)\in J$.