A non-symmetric polynomial whose square is symmetric

Over any field of characteristic different from \(2\), take \[ P(x,y)=x-y. \] It is not symmetric because \[ P(y,x)=y-x=-P(x,y)\ne P(x,y). \] But \[ P(x,y)^2=(x-y)^2 \] is symmetric, since interchanging \(x\) and \(y\) leaves it unchanged.

More generally, the Vandermonde polynomial \[ \Delta(x_1,\ldots,x_n)=\prod_{i<j}(x_i-x_j) \] is alternating: \[ \Delta(x_{\sigma(1)},\ldots,x_{\sigma(n)}) =\operatorname{sgn}(\sigma)\Delta(x_1,\ldots,x_n). \] Hence \(\Delta\) is nonsymmetric when the characteristic is not \(2\), while \[ \Delta^2=\prod_{i<j}(x_i-x_j)^2 \] is invariant under every permutation of the variables and therefore symmetric.