Polya and pre-Polya Groups in Even Dihedral Number Fields
27 APR 2021
16:30 - 18:30
For a number field K with the ideal class group Cl(K), Polya group of K is the subgroup Po(K) of Cl(K) generated by the classes of Ostrowski ideals Π_q (K), where q≥1 is a prime power integer and Π_q (K) denotes the product of all maximal ideals of K with norm q. K is called a Polya field, Whenever Po(K) is trivial. Polya fields are a generalization of PID (class number one) number fields, and classically they are defined in terms of regular bases for rings of integer valued polynomials due to George Polya. For Galois number fields K investigating on Polya-ness can be expressible in terms of the action of the Galois group on the ideal class group: Po(K) and the subgroup of Cl(K) generated by the strongly ambiguous ideal classes coincide. In particular, Zantema (whose paper is a great contribution in this subject) showed that in the Galois case, Polya groups are controllable part of ideal class groups throughout Galois cohomology and ramification. Beside, investigating on Polya groups in the non-Galois number fields (the more difficult situation), Chabert introduced the notion of pre-Polya group Po(-)_nr, which is a generalization of the pre-Polya condition, duo to Zantema.
The first part of my talk would be about some results of a joint work with Ali Rajaei, where using Zantema's result and the arithmetic in ramification theory, we found some results on Polya groups of dihedral extensions of Q of order 2l for l an odd prime. In the second part, I'll talk about my recently results on the pre-Polya group of a D_n-field K, for n≥4 an even integer, where D_n denotes the dihedral group of order 2n.
This is the first joint webinar organized by FGC-Istanbul and IPM-Tehran.
Zoom Meeting Link:
Meeting ID: 929 970 0405