Article
| Title: | On the irreducible factors of a polynomial over a valued field (English) |
| Author: | Jakhar, Anuj |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 2 |
| Year: | 2024 |
| Pages: | 367-375 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We explicitly provide numbers $d$, $e$ such that each irreducible factor of a polynomial $f(x)$ with integer coefficients has a degree greater than or equal to $d$ and $f(x)$ can have at most $e$ irreducible factors over the field of rational numbers. Moreover, we prove our result in a more general setup for polynomials with coefficients from the valuation ring of an arbitrary valued field. (English) |
| Keyword: | irreducibility |
| Keyword: | Eisenstein criterion |
| Keyword: | polynomial |
| MSC: | 11R09 |
| MSC: | 12E05 |
| MSC: | 12J10 |
| DOI: | 10.21136/CMJ.2024.0451-22 |
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| Date available: | 2024-07-10T14:49:11Z |
| Last updated: | 2024-07-15 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152445 |
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| Reference: | [10] Jhorar, B., Khanduja, S. K.: Reformulation of Hensel's lemma and extension of a theorem of Ore.Manuscr. Math. 151 (2016), 223-241. Zbl 1351.12002, MR 3532244, 10.1007/s00229-016-0829-z |
| Reference: | [11] 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: | [12] Murty, M. Ram: Prime numbers and irreducible polynomials.Am. Math. Mon. 109 (2002), 452-458. Zbl 1053.11020, MR 1901498, 10.1080/00029890.2002.11919872 |
| Reference: | [13] Schönemann, T.: Von denjenigen Moduln, welche Potenzen von Primzahlen sind.J. Reine Angew. Math. 32 (1846), 93-105 German. MR 1578516, 10.1515/crll.1846.32.93 |
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