On the Bring normal form of a quintic equation in characteristic 5 (Q1193279)

Let \(k\) be a field and \(f=x^ 5-a_ 1x^ 4+a_ 2x^ 3-a_ 3x^ 2+a_ 4x-a_ 5\) be a separable irreducible polynomial in \(k[x]\) with roots \(\alpha_ 0,\alpha_ 1,\dots,\alpha_ 4\) in an algebraic closure \(\bar k\) of \(k\) and Galois group \(G\). Put \(\alpha=\alpha_ 0\). The author poses the following Question: Can we find a \(\beta\in k(\alpha)\) with \(k(\beta)=k(\alpha)\) such that \(\beta\) is a root of a polynomial \(q=x^ 5+b_ 4x-b_ 5\) with \(b_ 4\), \(b_ 5\in k\). \(g\) is then called Bring normal form of \(f\). The author answers this question. The answer is affirmative if characteristic of \(k\neq 5\) after extending \(k\) to a field \(k'\) where \(k'\) is solvable over \(k\) and \(f\) remains irreducible over \(k'\). If characteristic \(k=5\), the answer is affirmative if and only if \(G\) is solvable.