Geodesic
On the irreducible factors of a polynomial over a valued field
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 367-375
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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.
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.
DOI : 10.21136/CMJ.2024.0451-22
Classification : 11R09, 12E05, 12J10
Keywords: irreducibility; Eisenstein criterion; polynomial 
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Jakhar, Anuj. On the irreducible factors of a polynomial over a valued field. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 367-375. doi: 10.21136/CMJ.2024.0451-22

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