Isotropy and full splitting pattern of quasilinear \(p\)-forms. (English) Zbl 07897474
Summary: For a quasilinear \(p\)-form defined over a field \(F\) of characteristic \(p > 0\), we prove that its defect over the field \(F(\sqrt[p^{n_1}]{a_1}, \dots, \sqrt[p^{n_r}]{a_r})\) is equal to its defect over the field \(F(\sqrt[p]{a_1}, \dots, \sqrt[p]{a_r})\), strengthening a result of Hoffmann from 2004. We also compute the full splitting pattern of some families of quasilinear \(p\)-forms.
MSC:
| 11E04 | Quadratic forms over general fields |
| 11E81 | Algebraic theory of quadratic forms; Witt groups and rings |
Keywords:
quasilinear \(p\)-forms; quadratic forms; finite characteristic; isotropy; full splitting patternReferences:
| [1] | Chapman, Adam, Linkage of sets of cyclic algebras, J. Pure Appl. Algebra, 226, 3, Article 106855 pp., 2022 · Zbl 1478.16011 |
| [2] | Gille, Philippe; Szamuely, Tamás, Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics, 2006, Cambridge University Press · Zbl 1137.12001 |
| [3] | Hoffmann, Detlev W., Diagonal forms of degree p in characteristic p, (Algebraic and Arithmetic Theory of Quadratic Forms. Algebraic and Arithmetic Theory of Quadratic Forms, Contemp. Math., vol. 344, 2004), 135-183 · Zbl 1074.11023 |
| [4] | Hoffmann, Detlev W., Splitting of quadratic Pfister forms over purely inseparable extensions in characteristic 2, J. Algebra, 596, 311-327, 2022 · Zbl 1487.11035 |
| [5] | Hoffmann, Detlev W.; Laghribi, Ahmed, Quadratic forms and Pfister neighbors in characteristic 2, Trans. Am. Math. Soc., 356, 10, 4019-4053, 02 2004 · Zbl 1116.11020 |
| [6] | Hoffmann, Detlev W.; Laghribi, Ahmed, Isotropy of quadratic forms over the function field of a quadric in characteristic 2, J. Algebra, 295, 2, 362-386, 2006 · Zbl 1138.11012 |
| [7] | Knebusch, Manfred, Generic splitting of quadratic forms, I, Proc. Lond. Math. Soc., s3-33, 1, 65-93, 1976 · Zbl 0351.15016 |
| [8] | Knebusch, Manfred, Generic splitting of quadratic forms, II, Proc. Lond. Math. Soc., s3-34, 01 1977 · Zbl 0359.15013 |
| [9] | Laghribi, Ahmed, On the generic splitting of quadratic forms in characteristic 2, Math. Z., 240, 4, 711-730, 2002 · Zbl 1007.11017 |
| [10] | Laghribi, Ahmed, On splitting of totally singular quadratic forms, Rend. Circ. Mat. Palermo, 53, 325-336, 2004 · Zbl 1097.11017 |
| [11] | Laghribi, Ahmed; Mukhija, Diksha, The behavior of singular quadratic forms under purely inseparable extensions, J. Algebra, 632, 31-61, 2023 · Zbl 1531.11035 |
| [12] | Pickert, Günter, Inseparable Körpererweiterungen, Math. Z., 52, 81-136, 1950 · Zbl 0034.31002 |
| [13] | Scully, Stephen, Rational maps between quasilinear hypersurfaces, Compos. Math., 149, 3, 333-355, 2013 · Zbl 1315.11025 |
| [14] | Scully, Stephen, Hoffmann’s conjecture for totally singular forms of prime degree, Algebra Number Theory, 10, 5, 1091-1132, 2016 · Zbl 1350.11049 |
| [15] | Scully, Stephen, On the splitting of quasilinear p-forms, J. Reine Angew. Math., 2016, 713, 49-83, 2016 · Zbl 1343.11047 |
| [16] | Weisfeld, Morris, Purely inseparable extensions and higher derivations, Trans. Am. Math. Soc., 116, 435-449, 1965 · Zbl 0136.32004 |
| [17] | Zemková, Kristýna, Equivalence relations of quadratic forms in characteristic 2 and quasilinear p-forms, 2022, Technische Universität Dortmund, PhD thesis · Zbl 1485.11079 |
| [18] | Zemková, Kristýna, Isotropy of quadratic forms over function fields in characteristic 2, Commun. Algebra, 2024, in press · Zbl 1485.11079 |
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