Real fields and repeated radical extensions (Q1270358)

A field extension \(L/K\) is a radical extension if \(L = K(\alpha)\), where \(\alpha \in L\) and \(\alpha ^n \in K\) for some positive integer \(n\). An extension \(L/K\) is a repeated radical extension if there exists a chain \(K = L _0 \subseteq L _1 \subseteq \ldots \subseteq L _r = L\), where \(L _i / L _{i-1}\) is a radical extension for \(1 \leq i \leq r\). It is well known that intermediate fields of repeated radical extensions need not themselves be repeated radical extensions of the ground field. In fact the authors include a simplified proof of the following result: if \(K\) is any subfield of the real numbers \({\mathbb R}\) and \(f \in K[x]\) is irreducible, splits over \({\mathbb R}\) and any one of the roots of \(f\) lies in a real repeated radical extension of \({\mathbb Q}\), then \(\deg f\) must be a power of 2. Further, the authors prove this result for \(K\) quasireal, that is, \(\text{char } K = 0\) and \(\pm 1\) are the only roots of unity contained in \(K\). The main purpose of the paper under review is to show that in certain cases intermediate fields of repeated radical extensions are themselves repeated radical extensions. It is proved that if \(Q\) is a real field and \(L/Q\) is a repeated radical extension with \([L : Q]\) odd, then any intermediate field \(Q \subseteq K \subseteq L\) is a repeated radical extension of \(Q\). It is also shown that the condition \(Q\) real cannot be dropped. For the case \([L : Q]\) a power of 2, we have the same result without the hypothesis of \(Q\) being real; it is enough that its characteristic be different from 2. It is also proved for a real field \(Q\) and \(f \in Q[x]\) irreducible of odd degree, that, if \(f\) has some root \(\alpha\) in a real repeated radical extension of \(Q\), then \(\alpha\) is the only real root of \(f\). At the end of the paper, several interesting examples and remarks are given. For instance, it is shown, with an example, that if \(Q\) is a real field, \(f \in Q[x]\) is irreducible of degree \(n\) and \(n\) is not necessarily either odd or a power of 2, then the number of real roots of \(f\) is not at most the 2-part of \(n\), as it might be suggested by the results for the cases \(n\) odd and \(n\) a power of 2. A polynomial of degree 6 is given having 4 real roots.