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On the annihilating ideal for trace forms

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On the annihilating ideal for trace forms JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX MARTIN EPKENHANS On the annihilating ideal for trace forms Journal de Théorie des Nombres de Bordeaux, tome 15, no 1 (2003), p. 115-124. <http://www.numdam.org/item?id=JTNB_2003__15_1_115_0> © Université Bordeaux 1, 2003, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/legal.php). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 115- On the annihilating ideal for trace forms par MARTIN EPKENHANS RÉSUMÉ. Npus donnons plusieurs exemples de familles de formes trace dont l’idéal annulateur dans Z[X] est principal. Nous mon- trons aussi qu’en général, cet idéal n’est pas principal. ABSTRACT. We give several examples of classes of trace forms for which the ideal of annihilating polynomials is principal. We prove, that in general, the annihilating ideal is not a principal ideal. 1. Introduction My talk given at the 20th Journees Arithmétiques at Limoges in 1997 concludes with a question on the injectivity of a certain map defined in the context of Burnside rings and trace forms. Now we are able to give an affirmative answer. Theorem 6 enables us to reduce questions on trace forms to corresponding questions of trace forms of 2-groups. Let K be a field of characteristic different from 2. Since the Witt ring W(K) over K is an integral ring we may consider polynomials in Z[X] evaluated at an element 0 of W(K). We say a polynomial p(X) E Z[X] annihilates 0 = 0 in W (K). Definition 1. Let M be any class of quadratic forms. Then the annihilat- ing ideal 1M of M is defined

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