CERAMATHS - DMATHS seminar: presentation by Mohammed Taous

The weekly seminar of the Department of Mathematics will host, on Thursday, November 24, Mohammed Taous (Université Moulay Ismaïl,  Meknes, Morocco), for the following talk:

The Pólya group of certain number bodies and the capitulation problem

The Pólya group $\mathcal{P}_O(K)$ of a number field K is the subgroup of $\mathrm{C}_K$, the class group of K, generated by the classes of the products of prime ideals having the same absolute norm.  When $\mathcal{P}_O(K)$ is trivial, the body K is called a Pólya body. In this talk, our goal is to give the relation that exists between  $\mathcal{P}_O(K)$ and  $athcal{P}_O(L)$ such that K/L is an unbranched extension and study in detail the case where $K=\mathb{Q}(\sqrt{ d_{1}},\sqrt{ d_{2}})$, $L=mathb{Q}(\sqrt{d_{3}})$, with $d_i$ are integers without square factors such that $(d_{1}, d_{2})=1$, $d_1$ or $d_2\equiv1\pmod4$, $d_3=d_1d_2$ divisible by a prime congruent to $3 \pmod 4$ or the norm of the fundamental unit of $L$ is negative, which allows us and with the help of the capitulation problem to determine the groups of P capitulation to determine the Pólya groups of real biquadratic number bodies K. We then deduce the K bodies that are de  Pólya and the structure of the first cohomology group of K units.