Power bases in dihedral quartic fields
The present paper is a very nice contribution to the theory of power integral bases in quartic fields. Let \(L\) be a dihedral quartic field with quadratic subfield \(k\), with discriminants \(d_L,d_k\), respectively. Let \(\{1,\omega\}\) be the usual integral basis of \(k\). Then \(L=\mathbb{Q}(\vartheta)\) where \(\vartheta^2=a+b\omega, a,b\in Z\). Set \(\nu=N_{k/\mathbb{Q}}(\vartheta^2)\) and denote by \(\tau\) the element of order four of the Galois group of \(L\). The main result is the following: Let \(\alpha=x+y\vartheta (x,y\in k)\), \(C=N_{k/\mathbb{Q}}(2y)\) and \(A=(x-x^{\tau})^2- \text{Tr}_{k/\mathbb{Q}}(y^2\vartheta^2)\). Then \(\alpha\) generates a power integral basis in \(L\) if and only if \(\alpha\) is integer in \(L\), \(\nu d_k^2C^2=d_L\) and \(4A^2=C^2\nu\pm 4d_k\) with appropriate sign. As a consequence, the author shows, that if \(L\) has a power integral basis, then \(d_L\pm 4d_k^3\) is a square. Moreover, for mixed dihedral quartic fields \(L\) having power integral bases \(|d_L|\leq 4d_k^3\) holds. The last statement is interesting also because it allows one to describe all mixed dihedral quartic fields \(L\) having power integral bases and containing a quadratic subfield \(k\). This was recently done by \textit{Nyul} and the reviewer [Computing all monogeneous mixed dihedral quartic extensions of a quadratic field, Proc. XXI Journées Arithmetiques, J. Théorie Nombres Bordeaux (to appear)].
scientific article; zbMATH DE number 1310087