Let \(K\) be a field and \(G\) a finite group. A monic separable polynomial \(g(t_1, \dots , t_m, X) \in K(t_1, \dots , t_m)[X]\) is said to be generic for \(G\) over \(K\) if
(1) The Galois group of \(g\) as a polynomial in \(X\) over \(K(t_1, \dots , t_m)\) is \(G\), and
(2) If \(L\) is an infinite field containing \(K\) and \(N/L\) is a Galois field extension with group \(G\), then there exist \(\lambda_1, \dots , \lambda_m \in L\) such that \(N\) is the splitting field of \(g(\lambda_1, \dots , \lambda_m, X)\) over \(L\).
The polynomial \(g\) is descent-generic if it satisfies (1) and the stronger property
(2’) If \(L\) is an infinite field containing \(K\) and \(N/L\) is a Galois field extension with group \(H \leq G\), then there exist \(\lambda_1, \dots , \lambda_m \in L\) such that \(N\) is the splitting field of \(g(\lambda_1, \dots , \lambda_mX)\) over \(L\).
This article shows that conditions (2) and (2’) are equivalent. Thus if \(g \in K(t_1, \dots , t_m)[X]\) is a generic polynomial in the sense that every Galois extension \(N/L\) of infinite fields with group \(G\) and \(K \leq L\) is given by a specialization of \(g\), then also every Galois extension whose group is a subgroup of \(G\) arises in this way.