Explicit integral basis of \(\mathbb{Q}(\sqrt[p_1 p_2]{a})\) (Q2161333)

Let \(K\) be an algebraic number field and let \(\mathbb{Z}_K\) denote its ring of algebraic integers. It is a classical problem to determine an integral basis of \(K\), that is a \(\mathbb{Z}\)-module basis of \(\mathbb{Z}_K\). The paper solves this problem in case \(K\) is generated by a root \(\theta\) of an irreducible integral polynomial of the form \(X^{p_1p_2}-a\) with primes \(p_1,p_2\). The actual results are too long to be included in the review, one of the main results takes up an entire page. For example, using the notation introduced above, the following corollary can be obtained. If \(a\) is square-free and \(a^{p_i - 1}\) is not congruent to \(1\) modulo \(p_i^2\) for \(i = 1,2\), then \(\{1, \theta, \dots , \theta^{p_1p_2 - 1}\}\) is an integral basis.