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http://hdl.handle.net/11320/21008Pełny rekord metadanych
| Pole DC | Wartość | Język |
|---|---|---|
| dc.contributor.author | Okazaki, Hiroyuki | - |
| dc.contributor.author | Shidama, Yasunari | - |
| dc.date.accessioned | 2026-09-14T11:48:10Z | - |
| dc.date.available | 2026-09-14T11:48:10Z | - |
| dc.date.issued | 2008 | - |
| dc.identifier.citation | Formalized Mathematics, Volume 16, Issue 2, 2008, Pages 103-107 | pl |
| dc.identifier.issn | 1426-2630 | - |
| dc.identifier.uri | http://hdl.handle.net/11320/21008 | - |
| dc.description.abstract | In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if p is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ∗ = Z/pZ\{0} is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/p∗. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ∗ and prove it is cyclic. | pl |
| dc.language.iso | en | pl |
| dc.publisher | University of Białystok | pl |
| dc.rights.uri | https://creativecommons.org/licenses/by-sa/4.0/ | - |
| dc.title | Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ | pl |
| dc.type | Article | pl |
| dc.rights.holder | © 2009 Hiroyuki Okazaki, Yasunari Shidama, published by University of Białystok | pl |
| dc.rights.holder | This work is licensed under the Creative Commons License | pl |
| dc.identifier.doi | 10.2478/v10037-008-0015-1 | - |
| dc.description.Affiliation | Hiroyuki Okazaki - Shinshu University Nagano, Japan | pl |
| dc.description.Affiliation | Yasunari Shidama - Shinshu University Nagano, Japan | pl |
| dc.description.references | Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990. | pl |
| dc.description.references | Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990. | pl |
| dc.description.references | Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990. | pl |
| dc.description.references | Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finitesequences. Formalized Mathematics, 1(1):107–114, 1990. | pl |
| dc.description.references | Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996. | pl |
| dc.description.references | Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990. | pl |
| dc.description.references | Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990. | pl |
| dc.description.references | Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. | pl |
| dc.description.references | Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990. | pl |
| dc.description.references | Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321–328, 1990. | pl |
| dc.description.references | Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573–577, 1997. | pl |
| dc.description.references | Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179–186, 2004. | pl |
| dc.description.references | Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990. | pl |
| dc.description.references | Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990. | pl |
| dc.description.references | Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990. | pl |
| dc.description.references | Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001. | pl |
| dc.description.references | Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001. | pl |
| dc.description.references | Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1):49–58, 2004. | pl |
| dc.description.references | Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001. | pl |
| dc.description.references | Christoph Schwarzweller. The ring of integers, euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999. | pl |
| dc.description.references | Christoph Schwarzweller and Agnieszka Rowińska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135–142, 2006. | pl |
| dc.description.references | Christoph Schwarzweller and Andrzej Trybulec. The evaluation of multivariate polynomials. Formalized Mathematics, 9(2):331–338, 2001. | pl |
| dc.description.references | Dariusz Surowik. Cyclic groups and some of their properties– part I. Formalized Mathematics, 2(5):623–627, 1991. | pl |
| dc.description.references | Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990. | pl |
| dc.description.references | Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15–22, 1993. | pl |
| dc.description.references | Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990. | pl |
| dc.description.references | Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990. | pl |
| dc.description.references | Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990. | pl |
| dc.description.references | Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991. | pl |
| dc.description.references | Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990. | pl |
| dc.description.references | Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990. | pl |
| dc.description.references | Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990. | pl |
| dc.identifier.eissn | 1898-9934 | - |
| dc.description.firstpage | 103 | pl |
| dc.description.lastpage | 107 | pl |
| dc.identifier.citation2 | Formalized Mathematics | pl |
| Występuje w kolekcji(ach): | Formalized Mathematics, 2008, Volume 16, Issue 2 | |
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