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Polynomials Irreducible by Eisenstein's Criterion
Applicable Algebra in Engineering, Communication and Computing, 2003Let \(0 ...
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Irreducible polynomials and barker sequences
ACM Communications in Computer Algebra, 2007A Barker sequence is a finite sequence a o , ..., a n -1 , each term ±1, for which every sum Σ i a i a i ...
Peter B. Borwein +2 more
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The American Mathematical Monthly, 1935
(1935). Irreducibility of Polynomials. The American Mathematical Monthly: Vol. 42, No. 6, pp. 369-381.
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(1935). Irreducibility of Polynomials. The American Mathematical Monthly: Vol. 42, No. 6, pp. 369-381.
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Finding Irreducible and Primitive Polynomials
Applicable Algebra in Engineering, Communication and Computing, 1992The paper presents new fast constructions of irreducible and primitive polynomials. It contains the following main results: 1. For any \(N \in \mathbb{N}\) one can construct an irreducible polynomial of degree \(n = N + O (N \exp (-( \log \log N)^{1/2-\varepsilon}))\) over \(GF(p)\) in time \((p \log N)^{O(1)}\). 2. For sufficiently large \(Q\) one can
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Computational Aspects of Irreducible Polynomials
Computational Mathematics and Mathematical Physics, 2020In this paper, an upper bound for the height of polynomial divisors of a given polynomial belonging to \(\mathbb Z[x]\) is computed, which is better than an already known bound. The author also gives constructive examples of a class of irreducible polynomials over discrete valued fields using Newton's polygon.
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Mutual Irreducibility of Certain Polynomials
2004In a recent paper, Tsaban and Vishne [4] introduce linear transformation shift registers (TSRs) which generate sequences by an entire word with each iteration. The authors recently [1] proved that over \(\mathbb{F}_2\), irreducible TSRs occur in pairs. Now the results are generalized and extended for arbitrary finite fields. This aids in the search for
Michael Dewar, Daniel Panario
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1993
Let F be a polynomial over an integral domainR, \( F \in R\left[ {\vec X} \right]\). As with rational integers, we say that F is reducible if there exist polynomialsG,\( H \in R\left[ {\vec X} \right]\),neither of which is inR, such that \( F = G \cdot H\).Otherwise,P is said to be irreducible or prime.
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Let F be a polynomial over an integral domainR, \( F \in R\left[ {\vec X} \right]\). As with rational integers, we say that F is reducible if there exist polynomialsG,\( H \in R\left[ {\vec X} \right]\),neither of which is inR, such that \( F = G \cdot H\).Otherwise,P is said to be irreducible or prime.
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Polynomials and Irreducibility
2018In this chapter, we present facts on zeros of polynomials and discuss some basic methods to decide whether a polynomial is irreducible or reducible, including Gauss’ lemma, the reduction of polynomials modulo prime numbers ((irreducibility over finite fields), and Eisenstein’s criterion.
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On the Irreducibility of Certain Classes of Polynomials
American Journal of Mathematics, 1939Verf. beweist die Irreduzibilität der Polynome \[ \frac{g_0}{d_0} + g_1 \frac{x^r}{d_1(s-t)!} + g_2 \frac{x^{2r}}{d_2(2s-t)!} + \cdots + g_n \frac{x^{nr}}{d_n(ns-t)!} \] im Körper der rationalen Zahlen. Hierin sind \(n, r, s, t\) positive ganze rationale Zahlen mit \(t\le ns - 2\). Für \(r\ge 2\) muß \(n\ge 2\) sein. Für \(\nu s - t\le 0\) ist \((\nu s
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