Joubert's theorem fails in characteristic 2
Let \(L/K\) be a separable field extension of degree \(6\). A theorem of P. Joubert proved in 1867 asserts that if \(\mathrm{char}(K)\) \(\neq 2\), then \(L\) is generated over \(K\) by an element whose minimal polynomial over \(K\) is of the form \(x^6 + ax^4 + bx^2 + cx + d\) for some \(a, b ,c, d \in K\). In this paper, it is shown that the above result does not hold in general when \(\mathrm{char}(K)\) is \(2\) and in this case it holds when either \(K\) is \(C_{1}\)-field or there exists an intermediate field of degree \(3\) over \(K\). In fact the author gives an elegant construction of a class of separable field extensions \(L_n/K_n\) having characteristic \(2\) of degree \(n = 2\cdot 3^m\) for any \(m \geq 1\) such that there is no \(y \in L_n - K_n\) for which the minimal polynomial of \(y\) over \(K_n\) is of the form \(x^n + \sum\limits_{i=0}^{n-1}a_ix^i\) with \(a_{n-1} = a_{n-3} = 0\).
scientific article; zbMATH DE number 6365752