Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ*

Open access

Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ*

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 Z/pZ 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/pZ*. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ* and prove it is cyclic.

MML identifier: INT 7, version: 7.8.10 4.99.1005

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[5] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.

[6] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.

[7] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.

[8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

[10] Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.

[11] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573-577, 1997.

[12] Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004.

[13] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.

[14] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.

[15] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.

[16] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.

[17] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.

[18] Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1):49-58, 2004.

[19] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.

[20] Christoph Schwarzweller. The ring of integers, euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.

[21] Christoph Schwarzweller and Agnieszka Rowińska-Schwarzweller. Schur's theorem on the stability of networks. Formalized Mathematics, 14(4):135-142, 2006.

[22] Christoph Schwarzweller and Andrzej Trybulec. The evaluation of multivariate polynomials. Formalized Mathematics, 9(2):331-338, 2001.

[23] Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.

[24] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.

[25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.

[26] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15-22, 1993.

[27] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

[28] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.

[29] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

[30] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.

[31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

Formalized Mathematics

(a computer assisted approach)

Journal Information


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Target Group

researchers in the fields of formal methods and computer-checked mathematics

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 25 25 7
PDF Downloads 8 8 4