The index of nonmonic polynomials
Let \(R \subset \mathcal R\) be integral domains and \(P = a_0 x^n + \dots +a_n\) a polynomial over \(R\). Put \(P_0=1\) and \(P_i=a_0 x^i + \dots + a_i\) for \(1 \leq i \leq n-1\), assume that \(\theta \in \mathcal R\) is a root of \(P\) and define \(R[P_k(\theta)]\), the invariant ring of \(P\), as the \(R\)-module generated by \(P_i(\theta),\;0 \leq i \leq n-1\); these ideas go back to R. Dedekind and A. Hurwitz. If \(P\) is irreducible, the discriminant of \(R[P_k(\theta)]\) over \(R\) equals \(\text{disc} (P)\). With these ideas the author generalizes the notion of index to nonintegral elements of an algebraic number field \(K\) and asks whether there exists some \(\theta \in K\) with index \(1\), i.e. for which \(\theta \) does \(\mathbb Z [P_k(\theta)]\) equal the ring of integers of \(K\). For cubic number fields this question is answered affirmatively (in contrast to the result for the classical index), for higher degrees such a \(\theta\) need no longer exist. Generalizing a result of \textit{M.-N. Gras} [J. Number Theory 23, 347--353 (1986; Zbl 0564.12008)] it is shown that almost all cyclic extensions of \(\mathbb Q\) do not have elements with index \(1\). Furthermore it is shown that with increasing conductor this index goes to \(\infty\).
scientific article; zbMATH DE number 1893618