Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring |
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Authors: | Nigel P Byott Boucha?¨b Soda?¨gui |
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Institution: | a Department of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, UK b Département de Mathématiques, Université de Valenciennes, Le Mont Houy, 59313 Valenciennes Cedex 9, France |
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Abstract: | Let k be a number field with ring of integers Ok, and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers ON of N determines a class in the locally free class group Cl(OkΓ]). We show that the set of classes in Cl(OkΓ]) realized in this way is the kernel of the augmentation homomorphism from Cl(OkΓ]) to the ideal class group Cl(Ok), provided that the ray class group of Ok for the modulus 4Ok 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 kΓ]. |
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Keywords: | 11R33 |
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