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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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Discriminants and the irreducibility of a class of polynomials
1986There has been some interest in finding irreducible polynomial of the type f(A(x)) for certain classes of linearized polynomials A(x) (see [1], [2], [3], [6],) over a finite field GF(pm). The main result of this paper proves the stronger result that there are no further irreducible cases of f(A(x)) fon an extended class that contains that of linearized
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On the reduction modulo p of an absolutely irreducible polynomial f (x, y)
Archiv Der Mathematik, 1997Umberto Zannier
exaly