| Title:
|
Prime ideal factorization in a number field via Newton polygons (English) |
| Author:
|
El Fadil, Lhoussain |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
2 |
| Year:
|
2021 |
| Pages:
|
529-543 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $K$ be a number field defined by an irreducible polynomial $F(X)\in \mathbb Z[X]$ and $\mathbb Z_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F(X)$ that guarantee the existence of exactly $r$ prime ideals of $\mathbb Z_K$ lying above $p$, where $\bar {F}(X)$ factors into powers of $r$ monic irreducible polynomials in $\mathbb F_p[X]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of $\mathbb Z_K$ lying above $p$. We further specify for every prime ideal of $\mathbb Z_K$ lying above $p$, the ramification index, the residue degree, and a $p$-generator. (English) |
| Keyword:
|
prime factorization |
| Keyword:
|
valuation |
| Keyword:
|
$\phi $-expansion |
| Keyword:
|
Newton polygon |
| MSC:
|
11S05 |
| MSC:
|
11Y05 |
| MSC:
|
11Y40 |
| idZBL:
|
07361083 |
| idMR:
|
MR4263184 |
| DOI:
|
10.21136/CMJ.2021.0516-19 |
| . |
| Date available:
|
2021-05-20T13:46:24Z |
| Last updated:
|
2023-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148919 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[5] Fadil, L. El, Montes, J., Nart, E.: Newton polygons and $p$-integral bases of quartic number fields.J. Algebra Appl. 11 (2012), Article ID 1250073, 33 pages. Zbl 1297.11134, MR 2959422, 10.1142/S0219498812500739 |
| Reference:
|
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| Reference:
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| Reference:
|
[8] Khanduja, S. K., Kumar, M.: Prolongations of valuations to finite extensions.Manuscr. Math. 131 (2010), 323-334. Zbl 1216.12007, MR 2592083, 10.1007/s00229-009-0320-1 |
| Reference:
|
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| . |
