When the Galois group of a polynomial is contained in $A_n$

Assume \(f\in K[x]\) is separable of degree \(n\) and \(\operatorname{char}K\ne2\). Let its roots in the splitting field be \[ \alpha_1,\dots,\alpha_n, \] and let \[ \delta=\prod_{i<j}(\alpha_i-\alpha_j). \] Then \[ \Delta(f)=\delta^2\in K. \] For \(\sigma\) in the Galois group, viewed as a permutation of the roots, \[ \sigma(\delta)=\operatorname{sgn}(\sigma)\delta. \] Therefore \[ G\subseteq A_n \iff \sigma(\delta)=\delta\quad\forall\sigma\in G \iff \delta\in K. \] Because \(\operatorname{char}K\ne2\) and \(\delta\ne0\), \[ \delta\in K \iff \Delta(f)=\delta^2\text{ is a square in }K. \] Hence \[ \boxed{G\subseteq A_n\iff \Delta(f)\in K^{\times2}}. \] The separability and characteristic assumptions are essential for this standard criterion in this form.