scholarly journals Superring of Polynomials over a Hyperring

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 902 ◽  
Author(s):  
Reza Ameri ◽  
Mansour Eyvazi ◽  
Sarka Hoskova-Mayerova
Keyword(s):  
Irreducible Polynomial ◽  
Basis Theorem

A Krasner hyperring (for short, a hyperring) is a generalization of a ring such that the addition is multivalued and the multiplication is as usual single valued and satisfies the usual ring properties. One of the important subjects in the theory of hyperrings is the study of polynomials over a hyperring. Recently, polynomials over hyperrings have been studied by Davvaz and Musavi, and they proved that polynomials over a hyperring constitute an additive-multiplicative hyperring that is a hyperstructure in which both addition and multiplication are multivalued and multiplication is distributive with respect to the addition. In this paper, we first show that the polynomials over a hyperring is not an additive-multiplicative hyperring, since the multiplication is not distributive with respect to addition; then, we study hyperideals of polynomials, such as prime and maximal hyperideals and prove that every principal hyperideal generated by an irreducible polynomial is maximal and Hilbert’s basis theorem holds for polynomials over a hyperring.

scholarly journals A basis theorem for free inverse semigroups

1977 ◽  
Vol 49 (1) ◽  
pp. 172-190 ◽  
Author(s):  
P.R Jones
Keyword(s):  
Inverse Semigroups ◽  
Basis Theorem

scholarly journals On purely periodic beta-expansions of Pisot numbers

2002 ◽  
Vol 166 ◽  
pp. 183-207 ◽  
Author(s):  
Yuki Sano
Keyword(s):  
Natural Extension ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Pisot Number ◽  
Fractal Boundary ◽  
Pisot Numbers ◽  
Dimensional Domain

AbstractWe characterize numbers having purely periodic β-expansions where β is a Pisot number satisfying a certain irreducible polynomial. The main tool of the proof is to construct a natural extension on a d-dimensional domain with a fractal boundary.

scholarly journals A generalization of the Burnside basis theorem

2014 ◽  
Vol 400 ◽  
pp. 8-16 ◽  
Author(s):  
Paul Apisa ◽  
Benjamin Klopsch
Keyword(s):  
Basis Theorem

scholarly journals On Monomorphisms and Subfields

2019 ◽  
Vol 27 (2) ◽  
pp. 133-137
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Isomorphic Copy ◽  
Field Extensions ◽  
The Third ◽  
Fourth Part ◽  
Roots Of Polynomials ◽  
Classical Proof ◽  
Do So

Summary This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/<p> as the desired field extension E [5], [3], [4]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/<p>. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/<p>, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ : F → F [X]/<p> and show that the translated polynomial ϕ (p) has a root over F [X]/<p>. Because F is not a subfield of F [X]/<p> we construct in this second part the field (E \ ϕF )∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/<p> and therefore consider F as a subfield of F [X]/<p>”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈 [X] = ∅ and 𝕉 ∩ 𝕉 [X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/<p> with the canonical monomorphism ϕ : F → F [X]/<p>. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root.

An S-box Design Using Irreducible Polynomial with Affine Transformation for Lightweight Cipher

2021 ◽  
pp. 214-227
Author(s):  
Muhammad Rana ◽  
Quazi Mamun ◽  
Rafiqul Islam
Keyword(s):  
Affine Transformation ◽  
Irreducible Polynomial ◽  
Lightweight Cipher

Irreducible Polynomial

2012 ◽  
Keyword(s):  
Irreducible Polynomial
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