Thus, denoting the ring of symmetric polynomials by Λn, the ring of symmetric and alternating polynomials is \Lambda_n[v_n], or more precisely \Lambda_n[v_n]/\langle v_n^2-\Delta\rangle, where \Delta=v_n^2 is a symmetric polynomial, the discriminant.
If 2 is not invertible, the situation is somewhat different, and one must use a different polynomial W_n, and obtains a different relation; see Romagny.