Sequences of residues in algebraic number fields (Q1343631)

Let \(K\) be an algebraic number field of degree \(d\) over the rationals. \(\Lambda_ K(n,m)\) denotes the least positive number such that for almost all prime ideals \(\mathfrak p\) of \(K\) there exists an integer \(\alpha\) and a unit \(u\) in \(K\) with the property that \(\alpha,\alpha + u,\dots,\alpha + (m - 1)u\) are all \(n\)-th power residues \(\bmod {\mathfrak p}\) with norm less than or equal to \(\Lambda_ K(n,m)^ d\). The author proves several theorems based on this definition. Here is an example: For any \(K\) of class number 1 and any integer \(n\), there exists an integer \(m\) such that \(\Lambda_ K(n,m) = \infty\).