Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

✍ Scribed by Nigel P. Byott; Bouchaı¨b Sodaı¨gui

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 290 KB
• Volume: 112
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.01.003

✦ Synopsis

Let k be a number field with ring of integers O k , and let be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to , the ring of integers O N of N determines a class in the locally free class group Cl(O k [ ]). We show that the set of classes in Cl(O k [ ]) realized in this way is the kernel of the augmentation homomorphism from Cl(O k [ ]) to the ideal class group Cl(O k ), provided that the ray class group of O k for the modulus 4O k has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367-378) on Galois module structure over a maximal order in k[ ].