INITIAL RAMIFICATION INDEX OF NONINVARIANT VALUATIONS ON FINITE DIMENSIONAL DIVISION ALGEBRAS
Let D be a division ring with centre K and dim, D< ? a valuation on K and v a noninvariant extension of ? to D. We define the initial ramfication index of v over ?, ?(v/ ?) .Let A be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to D with valuation ri...
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| Format: | Article |
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| Language: | English |
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University of Tehran
1997-09-01
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| Series: | Journal of Sciences, Islamic Republic of Iran |
| Online Access: | https://jsciences.ut.ac.ir/article_31257_15f75ef1d10234576e559849154d710f.pdf |
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| description | Let D be a division ring with centre K and dim, D< ? a valuation on K and v a
noninvariant extension of ? to D. We define the initial ramfication index of v over
?, ?(v/ ?) .Let A be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to D with valuation rings A , A ,…, A .
If B= A , it is shown that the following conditions are equivalent: (i) B is a finite A-module, (ii) B is a free A-module, (iii) [B/mB: A/m] = [D: k], (iv) e(v / ?) f(v / ?)= [D: K] and ? (v / ?)= e(v / ?). It is also proved that if ? (v/ ?) = e(v/ ?), and any of (i) - (iv) holds, then v is invariant |
| format | Article |
| id | doaj-art-41faf1f874ec47fd8c2f5c950bf0430a |
| institution | OA Journals |
| issn | 1016-1104 2345-6914 |
| language | English |
| publishDate | 1997-09-01 |
| publisher | University of Tehran |
| record_format | Article |
| series | Journal of Sciences, Islamic Republic of Iran |
| spelling | doaj-art-41faf1f874ec47fd8c2f5c950bf0430a2025-08-20T02:25:51ZengUniversity of TehranJournal of Sciences, Islamic Republic of Iran1016-11042345-69141997-09-018331257INITIAL RAMIFICATION INDEX OF NONINVARIANT VALUATIONS ON FINITE DIMENSIONAL DIVISION ALGEBRASLet D be a division ring with centre K and dim, D< ? a valuation on K and v a noninvariant extension of ? to D. We define the initial ramfication index of v over ?, ?(v/ ?) .Let A be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to D with valuation rings A , A ,…, A . If B= A , it is shown that the following conditions are equivalent: (i) B is a finite A-module, (ii) B is a free A-module, (iii) [B/mB: A/m] = [D: k], (iv) e(v / ?) f(v / ?)= [D: K] and ? (v / ?)= e(v / ?). It is also proved that if ? (v/ ?) = e(v/ ?), and any of (i) - (iv) holds, then v is invarianthttps://jsciences.ut.ac.ir/article_31257_15f75ef1d10234576e559849154d710f.pdf |
| spellingShingle | INITIAL RAMIFICATION INDEX OF
NONINVARIANT VALUATIONS ON FINITE
DIMENSIONAL DIVISION ALGEBRAS Journal of Sciences, Islamic Republic of Iran |
| title | INITIAL RAMIFICATION INDEX OF
NONINVARIANT VALUATIONS ON FINITE
DIMENSIONAL DIVISION ALGEBRAS |
| title_full | INITIAL RAMIFICATION INDEX OF
NONINVARIANT VALUATIONS ON FINITE
DIMENSIONAL DIVISION ALGEBRAS |
| title_fullStr | INITIAL RAMIFICATION INDEX OF
NONINVARIANT VALUATIONS ON FINITE
DIMENSIONAL DIVISION ALGEBRAS |
| title_full_unstemmed | INITIAL RAMIFICATION INDEX OF
NONINVARIANT VALUATIONS ON FINITE
DIMENSIONAL DIVISION ALGEBRAS |
| title_short | INITIAL RAMIFICATION INDEX OF
NONINVARIANT VALUATIONS ON FINITE
DIMENSIONAL DIVISION ALGEBRAS |
| title_sort | initial ramification index of noninvariant valuations on finite dimensional division algebras |
| url | https://jsciences.ut.ac.ir/article_31257_15f75ef1d10234576e559849154d710f.pdf |