\((A_n,S_n)\) realizations by polynomials -- on a question of Fried (Q1273221)

Let \(K\) be a field of characteristic 0 and \(f(X)\in K[X]\) be of degree \(n\). Let \(t\) be transcendental over \(K\), \(L\) a splitting field of \(f(X)-t\) over \(K(t)\). Let \(\widehat K\) be the algebraic closure of \(K\) in \(L\). \(A=G(L/K(t))\) is the arithmetic monodromy group of \(f\) and \(G=G(L/\widehat K(t))\) is the geometric monodromy group of \(f\). The author proves the following Theorem: Suppose that \(G\) consists of even permutations. Then \(n\) is odd and \(A\) consists of even permutations if and only if \((-1)^{{n-1\over 2}}n\) is a square in \(K\). In particular for \(K=\mathbb{Q},A\) consists of even permutations if and only if \(n\) is a square. -- This settles a question of M. Fried.