= Solution
Let $f(x)=a\prod_{i=1}^n(x-\alpha_i)$. Its <polynomial discriminant> is
$$
\Delta(f)=a^{2n-2}\prod_{i<j}(\alpha_i-\alpha_j)^2
=(-1)^{n(n-1)/2}a^{n-2}\operatorname{Res}(f,f').
$$
The squared <Vandermonde determinant> is a symmetric polynomial in the roots. By the <Fundamental theorem of symmetric polynomials>, it is a polynomial in their elementary symmetric functions, which are the coefficients of $f$ by <Vieta formulas>; hence $\Delta(f)\in K$. Equivalently, the <resultant> formula puts it directly in $K$.
For $f=x^4+rx+s$, a resultant calculation with $f'=4x^3+r$ gives
$$
\boxed{\Delta(f)=256s^3-27r^4}.
$$
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