Bases normales d'entiers relatives quadratiques. (Normal relative quadratic integral bases) (Q805667)

Let N/E be a finite Galois extension of number fields. Let \(\Delta\) be its Galois group, \({\mathcal O}_ N\), \({\mathcal O}_ E\) the integral rings. When is \({\mathcal O}_ N\) a free \({\mathcal O}_ E[\Delta]\)-module of rank 1? A necessary condition is that N/E is moderately ramified, but it is in general not sufficient. The case \(E={\mathbb{Q}}\) is solved by Fröhlich's school. In general, even in customary cases, the answer is sometimes negative. The author gives a positive answer for a broad class as follows. Problem \((E/K,\epsilon)\): Does there exist a field N above E, Galois over K, with \(\Delta\cong Gal(N/E)\), inducing a group extension \[ 1\quad \to \quad \Delta \quad \to \quad Gal(N/K)\quad \to \quad \Gamma \quad \to \quad 1 \] of class \(\epsilon\). If so, N/K is called a solution of the solvable problem \((E/K,\epsilon)\). Problem \([E/K,\epsilon]\): Is \((E/K,\epsilon)\) solvable; if so, does it admit a solution N/K s.t. one obtains a normal base of integers of N over E. The author gives an affirmative answer in case \(\Delta\) is a group of order 2 and E is a quadratic or biquadratic bicyclic extension of a number field K with class \(h(K)=1\). The results are constructive in that they can be applied to construct effectively by questioned normal bases.