Skew-Forms and Galois Theory
Let $L/K$ be a cyclic extension of degree $n = 2m$. It is known that the space $\mathrm{Alt}_K(L)$ of alternating $K$-bilinear forms (skew-forms) on $L$ decomposes into a direct sum of $K$-subspaces $A^{\sigma ^i}$ indexed by the elements of $\mathrm{Gal}(L/K) = \langle \sigma \rangle $. It is also...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.645/ |
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| author | Gupta, Ashish Mandal, Sugata |
| author_facet | Gupta, Ashish Mandal, Sugata |
| author_sort | Gupta, Ashish |
| collection | DOAJ |
| description | Let $L/K$ be a cyclic extension of degree $n = 2m$. It is known that the space $\mathrm{Alt}_K(L)$ of alternating $K$-bilinear forms (skew-forms) on $L$ decomposes into a direct sum of $K$-subspaces $A^{\sigma ^i}$ indexed by the elements of $\mathrm{Gal}(L/K) = \langle \sigma \rangle $. It is also known that the components $A^{\sigma ^i}$ can have nice constant-rank properties. We enhance and enrich these constant-rank results and show that the component $A^\sigma $ often decomposes directly into a sum of constant rank subspaces, that is, subspaces all of whose non-zero skew-forms have a fixed rank $r$. In particular, this is always true when $-1 \notin L^2$. As a result we deduce a decomposition of $\mathrm{Alt}_K(L)$ into subspaces of constant rank in several interesting situations. We also establish that a subspace of dimension $\frac{n}{2}$ all of whose nonzero skew-forms are non-degenerate can always be found in $A^{\sigma ^i}$ where $\sigma ^i$ has order divisible by $2$. |
| format | Article |
| id | doaj-art-da04436558ef44cc883cacf43883f9cf |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-da04436558ef44cc883cacf43883f9cf2025-02-07T11:23:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G111337134710.5802/crmath.64510.5802/crmath.645Skew-Forms and Galois TheoryGupta, Ashish0Mandal, Sugata1School of Mathematical Sciences, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah, West Bengal, Box: 711202, India.School of Mathematical Sciences, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah, West Bengal, Box: 711202, India.Let $L/K$ be a cyclic extension of degree $n = 2m$. It is known that the space $\mathrm{Alt}_K(L)$ of alternating $K$-bilinear forms (skew-forms) on $L$ decomposes into a direct sum of $K$-subspaces $A^{\sigma ^i}$ indexed by the elements of $\mathrm{Gal}(L/K) = \langle \sigma \rangle $. It is also known that the components $A^{\sigma ^i}$ can have nice constant-rank properties. We enhance and enrich these constant-rank results and show that the component $A^\sigma $ often decomposes directly into a sum of constant rank subspaces, that is, subspaces all of whose non-zero skew-forms have a fixed rank $r$. In particular, this is always true when $-1 \notin L^2$. As a result we deduce a decomposition of $\mathrm{Alt}_K(L)$ into subspaces of constant rank in several interesting situations. We also establish that a subspace of dimension $\frac{n}{2}$ all of whose nonzero skew-forms are non-degenerate can always be found in $A^{\sigma ^i}$ where $\sigma ^i$ has order divisible by $2$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.645/Alternating formskew-symmetric formconstant rank spaceGalois extension |
| spellingShingle | Gupta, Ashish Mandal, Sugata Skew-Forms and Galois Theory Comptes Rendus. Mathématique Alternating form skew-symmetric form constant rank space Galois extension |
| title | Skew-Forms and Galois Theory |
| title_full | Skew-Forms and Galois Theory |
| title_fullStr | Skew-Forms and Galois Theory |
| title_full_unstemmed | Skew-Forms and Galois Theory |
| title_short | Skew-Forms and Galois Theory |
| title_sort | skew forms and galois theory |
| topic | Alternating form skew-symmetric form constant rank space Galois extension |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.645/ |
| work_keys_str_mv | AT guptaashish skewformsandgaloistheory AT mandalsugata skewformsandgaloistheory |