Integral basis of the field \(\mathbb Q(\root n\of a)\) (Q585280)

Let \(\mathbb Q\) be the field of rational numbers, \(\mathbb Z\) the ring of rational integers, \(n,a\in\mathbb Z\), \(n\geq 2\), \((n,a)=1\). The main result of the paper under review is the following theorem: Let \(f(x)=x^n-a\) be an irreducible polynomial in \(\mathbb Z[x]\). Let \(\root n\of a\) be one of the roots of \(f(x)\) in \(\mathbb C\). Let \(n=\prod_{i=1}^k p_i^{s_i}\), \(a=\prod_{j=1}^l q_j^{t_j}\), where \(p_i\) \((i=1,2,\dots,k)\), \(q_j\) \((j=1,2,\dots,l)\) are distinct primes. Put \(n_i=n/p_i^{s_i}\) \((i=1,2,\dots,k)\). Now put \(g_0(x)=l\) and for any \(m\in\{1,2,\dots,n\}\) denote by \(g_n(x)\) a monic polynomial in \(\mathbb Z[x]\) satisfying \[ g_m(x)\equiv (x^{n_i}-a)^{[m/n_i]} x^{m-n_i[m/n_i]} \bmod p_i^{[[m/n_i]\gamma_i]+1}\quad (i=1,2,\dots,k), \] where \(\gamma_i=1/p_i^{s_i}\) when \(\text{ord}_{p_i}\left(a^{p_i^{s_i}}-1\right)=1\), \(\gamma_i=1/\varphi(p_i^{s_i})\) when \(\text{ord}_{p_i}\left(a^{p_i^{s_i}}-1\right)>1\). Then \[ \left\{\frac{g_m(a)}{\prod_{i=1}^k p_i^{[[m/n_i]\gamma_i]}\prod_{j=1}^l q_j^{[t_jm/n]}} \mid m=0,1,\dots, n-1\right\} \] is an integral basis of \(\mathbb Q(\root n\of a)\).