Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach)
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912974039416832 |
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| author | Bavula, V. V. |
| author_facet | Bavula, V. V. |
| contents | The aim of the paper is to introduce B-extensions which are the most symmetrical finite field extensions (a finite field extension $L/K$ is called a {\it B-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the algebra of differential operators ${\cal D} (L/K)$ on the $K$-algebra $L$ and the automorphism group $G(L/K):={\rm Aut}_{K-{\rm alg}}(L)$) and to obtain an analogue of the Galois Theory for B-extensions. Surprisingly, the class of B-extensions coincides with the class of {\it normal } finite field extensions. As a result, an analogue of the Galois Theory is obtained for normal field extensions. In particular, all Galois field extensions and all purely inseparable field extensions are B-extensions. Our approach is a ring theoretic (characteristic free) approach which is based on central simple algebras. In this approach, analogues of the Galois Correspondences (for subfields and normal subfields of $L$) are deduced from the Double Centralizer Theorem which is applied to the central simple algebra ${\rm End}_K(L)$ and subfields of B-extensions.
Since Galois finite field extensions are B-extensions, this approach gives a new conceptual (short) proofs of key results of the Galois Theory, see [2] for details. It also reveals that the `maximal symmetry' (of field extensions) is the essence of the classical Galois Theory and the analogue of the Galois Theory for normal field extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach) Bavula, V. V. Number Theory 11S20, 12F10, 16S32, 12F05, 12F15, 16G10 The aim of the paper is to introduce B-extensions which are the most symmetrical finite field extensions (a finite field extension $L/K$ is called a {\it B-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the algebra of differential operators ${\cal D} (L/K)$ on the $K$-algebra $L$ and the automorphism group $G(L/K):={\rm Aut}_{K-{\rm alg}}(L)$) and to obtain an analogue of the Galois Theory for B-extensions. Surprisingly, the class of B-extensions coincides with the class of {\it normal } finite field extensions. As a result, an analogue of the Galois Theory is obtained for normal field extensions. In particular, all Galois field extensions and all purely inseparable field extensions are B-extensions. Our approach is a ring theoretic (characteristic free) approach which is based on central simple algebras. In this approach, analogues of the Galois Correspondences (for subfields and normal subfields of $L$) are deduced from the Double Centralizer Theorem which is applied to the central simple algebra ${\rm End}_K(L)$ and subfields of B-extensions. Since Galois finite field extensions are B-extensions, this approach gives a new conceptual (short) proofs of key results of the Galois Theory, see [2] for details. It also reveals that the `maximal symmetry' (of field extensions) is the essence of the classical Galois Theory and the analogue of the Galois Theory for normal field extensions. |
| title | Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach) |
| topic | Number Theory 11S20, 12F10, 16S32, 12F05, 12F15, 16G10 |
| url | https://arxiv.org/abs/2509.01779 |