Explicit discriminant and integral basis of \({{\textbf{Q}}}(\root p^2 \of {a})\)

Abstract

Let \(K = {{\textbf{Q}}}(\theta )\) be an algebraic number field, where \(\theta \) is a root of the irreducible polynomial \(x^n - a \in {{\textbf{Q}}}[x]\) of degree n, and let \({\textbf {Z}}_K\) denote the ring of integers of K. In general, for arbitrary integers n and a, the explicit computation of the discriminant and an integral basis of K remains unresolved, except in special cases such as \(n = 4\), 8, 9, a prime p, or a product pq of distinct primes p and q. Most existing results assume either that a is square-free or that \(\gcd (a, n) = 1\). In this article, we address the case \(n = p^2\) for an odd prime p, without imposing any condition on a. Using the theory of Newton polygons of first and second order, we determine the exact power of each prime q dividing the index \([{\textbf {Z}}_K : {\textbf {Z}}[\theta ]]\). This allows us to construct explicitly a q-integral basis of K for each prime q, which in turn yields a complete integral basis of K. Several illustrative examples are provided. As an application, we show that if a is square-free and satisfies \(a^{p-1} \not \equiv 1 \pmod {p^2}\), then K admits a power basis; that is, \({\textbf {Z}}_K = {\textbf {Z}}[\theta ]\).