Let be a domain which is finitely generated over and integrally closed in its quotient field . Further, let be a finite extension field of . An -order in is a domain with quotient field which is integral over . -orders in of the type are called monogenic. It was proved by Győry [10] that for any given -order in there are at most finitely many -equivalence classes of with , where two elements of are called -equivalent if for some , . If the number of -equivalence classes of with is at least , we call times monogenic.
In this paper we study orders which are more than one time monogenic. Our first main result is that if is any finite extension of of degree , then there are only finitely many three times monogenic -orders in . Next, we define two special types of two times monogenic -orders, and show that there are extensions which have infinitely many orders of these types. Then under certain conditions imposed on the Galois group of the normal closure of over , we prove that has only finitely many two times monogenic -orders which are not of these types. Some immediate applications to canonical number systems are also mentioned.
@article{ASNSP_2013_5_12_2_467_0, author = {B\'erczes, Attila and Evertse, Jan-Hendrik and Gy\H{o}ry, K\'alm\'an}, title = {Multiply monogenic orders}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {467--497}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 12}, number = {2}, year = {2013}, mrnumber = {3114010}, zbl = {1319.11070}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2013_5_12_2_467_0/} }
TY - JOUR AU - Bérczes, Attila AU - Evertse, Jan-Hendrik AU - Győry, Kálmán TI - Multiply monogenic orders JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2013 SP - 467 EP - 497 VL - 12 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2013_5_12_2_467_0/ LA - en ID - ASNSP_2013_5_12_2_467_0 ER -
%0 Journal Article %A Bérczes, Attila %A Evertse, Jan-Hendrik %A Győry, Kálmán %T Multiply monogenic orders %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2013 %P 467-497 %V 12 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2013_5_12_2_467_0/ %G en %F ASNSP_2013_5_12_2_467_0
Bérczes, Attila; Evertse, Jan-Hendrik; Győry, Kálmán. Multiply monogenic orders. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 2, pp. 467-497. http://www.numdam.org/item/ASNSP_2013_5_12_2_467_0/
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