On solvable quintics \(X^ 5 + aX +b\) and \(X^ 5 + aX^ 2 +b\)
This is a very fine paper. Let \(\mathbb{Q}\) be the field of rational numbers. The authors first obtain explicit solutions of \(X^5+ aX^2+b\) when it is both irreducible and solvable. Let \(F(1)= \{f_1(X) = X^5+ aX+b: f_1(X)\) is irreducible and solvable with Galois group isomorphic to the Frobenius group \(F_{20}\}\) and \(D(1)= \{f_1(X) = X^5+ aX+b: f_1 (X)\) irreducible and solvable with Galois group isomorphic to \(D_5\), the dihedral group of order 10\}; \(F(2)\), \(D(2)\) are similarly defined with trinomials \(X^5+ aX^2+b\). If \(f_1,g_1 \in F(1)\) or \(D(1)\), \(f_1 \backsim g_1\) if there exists \(t\in \mathbb{Q}^*\) with \(a_1= at^{5-i}\), \(b_1= bt^5\). Similarly \(\backsim\) is defined in \(D(1)\), \(F(2)\), \(D(2)\). Let \({\mathcal F}(1)= F(1)/ \backsim\) and define \({\mathcal D} (1), {\mathcal F} (2), {\mathcal D} (2)\) similarly. The authors prove \[ \bigl|{\mathcal F} (1) \bigr|= \infty,\;\bigl|{\mathcal D} (1)\bigr|= \infty, \bigl|{\mathcal F} (2)\bigr|= 2,\bigl |{\mathcal D} (2) \bigr|=3. \]
scientific article; zbMATH DE number 944417