Kesiklikan Grup Unit dari Ring Z_n - Damanik, Dieny Ahda; Amri, Saiful; Hafnani, Hafnani (Syiah Kuala University, 2023)
The unit group of rings Z_n denoted by U_n is widely used in pure mathematics and applied mathematics. It makes the mathematics calculation in U_n more complicated as n gets bigger. However, when U_n is cyclic, the calculation is no longer matter. In previous research, it was shown that U_n is cyclic if and only if n = 2;4; p^k;2p^k for any odd prim p and any positive integer k. This statement is known as Gauss Theorem for primitive root. This paper aims to reprove The Gauss Theorem using various tools like divisibility in Z, group and ring, polynomials over integral domain and fields. It is stated by showing that U_ab is not cyclic where a,b 2, (a,b) = 1. Thus, the necessity for U_n to be cyclic, n must be of the form 2k; pk or 2pk. Furthermore, we find U_(2^k ) is cyclic if only if k 1,2. And then, proved that U_p is cyclic and using induction on k, we find that U_(p^k ) is cyclic. Finally, it is also found that U_(2p^k ) is cyclic so that Gauss Theorem is proven.