The Galois extensions induced by idempotents in a Galois algebra
Let B be a Galois algebra with Galois group G, Jg={b∈B|bx=g(x)b for all x∈B} for each g∈G, eg the central idempotent such that BJg=Beg, and eK=∑g∈K,eg≠1eg for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)(={g∈G|g(eK)=eK}) containing K and the normalizer N(K) of K...
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| Format: | Article |
| Language: | English |
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202007767 |
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| _version_ | 1832550146405040128 |
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| author | George Szeto Lianyong Xue |
| author_facet | George Szeto Lianyong Xue |
| author_sort | George Szeto |
| collection | DOAJ |
| description | Let B be a Galois algebra with Galois group G, Jg={b∈B|bx=g(x)b for all x∈B} for each g∈G, eg the central idempotent such that BJg=Beg, and eK=∑g∈K,eg≠1eg for a subgroup K of G. Then BeK is a Galois extension with the Galois group
G(eK)(={g∈G|g(eK)=eK}) containing K and the normalizer N(K) of K in G. An equivalence condition is also
given for G(eK)=N(K), and BeG is shown to be a direct sum of
all Bei generated by a minimal idempotent
ei. Moreover, a
characterization for a Galois extension B is shown in terms of
the Galois extension BeG
and B(1−eG). |
| format | Article |
| id | doaj-art-73eb842cf4f841e196d86fc421fd95b4 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-73eb842cf4f841e196d86fc421fd95b42025-02-03T06:07:35ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0129737538010.1155/S0161171202007767The Galois extensions induced by idempotents in a Galois algebraGeorge Szeto0Lianyong Xue1Department of Mathematics, Bradley University, Peoria 61625, IL, USADepartment of Mathematics, Bradley University, Peoria 61625, IL, USALet B be a Galois algebra with Galois group G, Jg={b∈B|bx=g(x)b for all x∈B} for each g∈G, eg the central idempotent such that BJg=Beg, and eK=∑g∈K,eg≠1eg for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)(={g∈G|g(eK)=eK}) containing K and the normalizer N(K) of K in G. An equivalence condition is also given for G(eK)=N(K), and BeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1−eG).http://dx.doi.org/10.1155/S0161171202007767 |
| spellingShingle | George Szeto Lianyong Xue The Galois extensions induced by idempotents in a Galois algebra International Journal of Mathematics and Mathematical Sciences |
| title | The Galois extensions induced by idempotents in a Galois algebra |
| title_full | The Galois extensions induced by idempotents in a Galois algebra |
| title_fullStr | The Galois extensions induced by idempotents in a Galois algebra |
| title_full_unstemmed | The Galois extensions induced by idempotents in a Galois algebra |
| title_short | The Galois extensions induced by idempotents in a Galois algebra |
| title_sort | galois extensions induced by idempotents in a galois algebra |
| url | http://dx.doi.org/10.1155/S0161171202007767 |
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