Kummer extensions with few roots of unity
Let \(K\) be an arbitrary field, \(\overline{K}\) an algebraic closure of \(K\), \(n\geq 1\) a natural number, and \(\mu_ n(\overline{K})=\{z|\) \(z\in\overline{K}\), \(z^ n=1\}\). A finite Kummer extension of \(K\) of exponent \(n\) with few (resp., many) roots of unity is an extension \(K(x_ 1,\dots,x_ n)\) of \(K\), where \(k\in\mathbb{N}^*\), \(x_ 1,\dots,x_ k\in\overline {K}^*\) are such that \(x_ i^ n\in K\) of all \(i\), \(1\leq i\leq k\), and \(\mu_ n(\overline{K})\cap K(x_ 1,\dots,x_ k)\subseteq \{1,-1\}\) (resp., \(\mu_ n(\overline{K})\subseteq K)\). The author shows that a classical result concerning the evaluation of the degree \({[K(x_ 1,\dots,x_ k):K]}\) holds equally for finite Kummer extensions of exponent \(n\) with few or with many roots of unity, if \(\text{Char}(K)\nmid n\). For such an extension \(K\subseteq K(x_ 1,\dots,x_ n)\) for which \({[K(x_ 1,\dots,x_ k):K]}=\prod_{1\leq i\leq k}[K(x_ i):K]\), it is shown that \(K(x_ 1,\dots,x_ k)=K(x_ 1+\dots+x_ k)\). Further, if \(K\) is an arbitrary field and \(n\) is a prime number other than \(\text{Char}(K)\), then any extension \(K\subseteq K(x_ 1,\dots,x_ k)\), where \(k\in \mathbb{N}^*\) and \(x_ 1,\dots,x_ k\in\overline{K}^*\) are such that \(x_ i^ n\in K\) for all \(i\), \(1\leq i\leq k\), is a finite Kummer extension of exponent \(n\) with few or with many roots of unity, and, consequently, the above results hold in this case.
scientific article; zbMATH DE number 96568
scientific article