How to prove following Generalization of Hilbert Nullstellansatz This question is from assignment 1 of my algebraic geometry course. For theory, I have been following my class notes.

Question: Let K be an arbitrary field, S be the set of all polynomials in $K[X_1, ..., X_n]$ which have no zeroes in $K^n$ and let A be an ideal in $K[X_1,...,X_n]$ . If $S\cap A =\varnothing$ , then show that $V_K(A)\neq \varnothing$.

Thoughts: $S\cap A=\varnothing$ implies that all elements of A have at least 1 zero in $K^n$. I have to prove that all elements of A have at least 1 common zero. But, why should all elements have at least one common zero? I am not sure what from the information given in the question would be useful to proceed foreward.

Can you please give some hints so that I could able to proceed to proving this question?

 A: By Nullstellensatz, if $V_k(A)=\emptyset$, then the radical ideal of $A$ (and hence $A$) corresponds to the entire space $K[X_1,\dots X_n]$, so that $1=1^k\in A$
However $1$ has not roots in $K^n$, so that $1\in S$. This means $S\cap A\neq \emptyset$, a contradiction.
A: If $K$ is algebraically closed, then this is the classical Hilbert's Nullstellensatz: $S$ consists of the constant non-zero polynomials, so $S \cap A = \emptyset$ if and only if $A$ is a proper ideal, if and only if (by the classical Nullstellensatz) $V_k(A) \neq \emptyset$. This is also the approach which @Federico Fallucca alludes to in their answer.
So what happens when $K$ is not algebraically closed? You start with a (WLOG finitely generated) ideal $A = (f_1, ..., f_k)$ for some $f_1, \ldots, f_k \in K[X_1, \ldots, X_n]$ and you know that all elements of $A$ have a root in $K^n$, and now you want to show that $f_1, \ldots, f_k$ also have a common root in $K^n$.
Let's think of $K = \mathbb{R}$ or $K = \mathbb{Q}$ for starters. In this case, you could observe that $f_1^2 + \ldots + f_k^2 \in A$ and hence has a root in $a \in K^n$. But then $f_1(a)^2 + \ldots + f_k(a)^2 = 0$ in $K$, which is only possible if $f_1(a) = \ldots = f_k(a) = 0$. So, we have found the desired common root of $f_1, \ldots, f_k$.
How would you generalise this observation to apply to an arbitrary field $K$?
