<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml"><p>In the [20], it had been proven that the Integers modulo <italic>p</italic>, in this article we shall refer as <italic>Z/pZ</italic>, constitutes a field if and only if <italic>Z/pZ</italic> is a prime. Then the prime modulo <italic>Z/pZ</italic> is an additive cyclic group and <italic>Z/pZ*</italic> = <italic>Z/pZ</italic>\{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 <italic>Z/pZ</italic>*. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group <italic>Z/pZ</italic>* and prove it is cyclic.</p><p>MML identifier: INT 7, version: 7.8.10 4.99.1005</p></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 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

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

Formalized Mathematics

Uniqueness of Factoring an Integer and Multiplicative Group 

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...

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

Online veröffentlicht: 20. März 2009

Seitenbereich: 103 - 107

DOI: https://doi.org/10.2478/v10037-008-0015-1

This content is open access.