Newton polygons and \(p\)-integral bases of quartic number fields
Consider a prime number \(p\) and a monic irreducible polynomial \(F(X)\in\mathbb{Z}[X]\). NEWLINENEWLINE\noindent Let \(F(X)\equiv \pi_1(X)^{\mu_1}\ldots\pi_r(X)^{\mu_r}\) modulo \(p\), where the \(\pi_i(X)\) are distinct and irreducible modulo \(p\). Then \(F(X)\) is said to be \(p\)-regular if it is regular with respect to \(\pi_1(X),\ldots,\pi_r(X)\). Suppose that \(F(X)\) is the minimal polynomial of \(\theta\) over \(\mathbb{Q}\) and let \(K=\mathbb{Q}(\theta)\). Using the results of \textit{Ö. Ore} [Acta Math. 44, 219--314 (1923; JFM 49.0698.04); Math. Ann. 99, 84--117 (1928; JFM 54.0191.02)], the authors determine efficiently a \(p\)-integral basis of \(K\). To illustrate the wide range of applicability of their result, they use it to obtain an explicit \(p\)-integral basis of an arbitrary quartic number field in terms of a defining equation.
scientific article; zbMATH DE number 6078482