Field extensions defined by power compositional polynomials
Let \(k>1\) and \(m>0\) be integers, \(F\) a field of characteristic \(0\), and \(f(x)\in F[x]\) an irreducible polynomial of degree \(km\). Let \(L = F(\alpha)\), where \(\alpha\) is a root of \(f(x)\). The aim of this paper (under review) is to study the following problem formulated in the light of the results of \textit{C. Awtrey} and \textit{P. Jakes} [Can. Math. Bull. 63, No. 3, 670--676 (2020; Zbl 1458.11173)], \textit{J. W. Jones} and \textit{D. P. Roberts} [J. Number Theory 128, No. 6, 1410--1429 (2008; Zbl 1140.11056)], \textit{M.-c. Kang} [Am. Math. Mon. 107, No. 3, 254--256 (2000; Zbl 0990.12001)]: when does there exist an irreducible polynomial \(g(x^k)\in F[x]\) such that \(L\) is isomorphic to \(F[x]/(g(x^k))\), and how do we construct such a polynomial \(g(x)\), if it exists? If \(F\) contains the \(k\)-th roots of unity, then Kummer theory provides an answer. More precisely, the author has shown using the classical result from Kummer theory that \(L/F\) can be defined by \(g(x^k)\) if and only if \(L/F\) has an automorphism \(\sigma\) of order \(k\) such that the fixed field of the cyclic group generated by \(\sigma\) contains the \(k\)-th roots of unity. Furthermore, he gave an algorithm, based on \(\sigma\), which allows the construction of \(g(x^k)\). This approach is not new; for example, the special case \(k=2\) was used in [\textit{J. W. Jones} and \textit{D. P. Roberts}, J. Number Theory 128, No. 6, 1410--1429 (2008; Zbl 1140.11056), Section 2] to construct irreducible even polynomials of degree 8 defining octic 2-adic fields with a quartic subfield. In particular, \(L/F\) is defined by an irreducible even polynomial in \(F[x]\) of degree \(2m\) if and only if \(L/F\) has an even number of automorphisms. However, the analogous statement is not necessarily true for \(k>2\). If \(L\) does not contain the \(k\)-th roots of unity, the main result is to explicitly describe the extensions \(L/F\) which are defined by irreducible polynomials of the form \(x^6 + ax^3 + b\). For this purpose, the author used \textit{M.-C. Kang}'s result [Am. Math. Mon. 107, No. 3, 254--256 (2000; Zbl 0990.12001)] to give a complete answer when \(k=3\) and \([L : F] = 6\), namely Kang supplied an elementary proof to produce a necessary and sufficient condition on the minimal polynomial of \(\alpha\) for \(L\) to be a radical extension of \(F\). The author notably constructs a polynomial \(x^6+ax^3+b\) defining \(L/F\), when it exists. As an application, he gives a simple method for determining the Galois group of an irreducible polynomial \(x^6+ax^3+b \in F[x]\). Examples are presented illustrating the application of the results obtained.
scientific article; zbMATH DE number 7492924
scientific article