Using Divisor Function and Euler Product Function in Abstract Algebra Concepts

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory. In this paper for the most part centered around number theory ideas which are utilized in different themes like group theory and ring theory, these speculations are extremely unique ideas to comprehend among this we might want to express our perspectives as far as number hypothesis/theory ideas, such as, to calculate some subgroups of a cyclic group, number of ideals, principal ideals of a ring and number of generators of a cyclic group as far as both regular procedure and number speculation/hypothesis thoughts.


Introduction
The theory of numbers is an area of mathematics which deals with the properties of whole and rational numbers. Analytic number theory is one of its branches, which involves study of arithmetical functions, their properties and the interrelationships that exist among these functions. In this paper I will introduce some of the three very important examples of arithmetical functions, as well as a concept of the possible operations we can use with them. There are four propositions which are mentioned in this paper and I have used the definitions of these arithmetical functions and some Lemmas which reflect their properties, in order to prove them.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
List Definition: A group G is called cyclic if there exists an element g in G such that G = <g> = {g n | n is an integer}. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains g is G itself suffices to show that G is cyclic.
Example1: If G is a cyclic group of order 12 then the number of subgroups of G. Solution: Here the number is 12 is finite and small number so to find out subgroup of these are very small task because the number of divisors of 12 are 1, 2, 3, 4, 6, and 12 so number of subgroups is 6.
If the number is large then to find out the number of subgroups is very difficult and time consuming process so to reduce this difficulty by using the following number theory concepts.
Definition: (Prime and Composite). An integer n > 1 is prime if it the only positive divisors of n are 1 and n. We call n composite if n is not prime.
The number 1 is neither prime nor composite. for n an integer is defined as the sum of the k th powers of the (positive integer) divisors of n, The notations d (n) (Hardy and Wright 1979, p. 239), v (n) (Ore 1988, p. 86), and The divisor function can also be generalized to Gaussian integers. The definition requires some care since in principle, there is ambiguity as to which of the four associates is chosen for each divisor. Spira (1961) defines the sum of divisors of a complex number Z by factoring Z into a product of powers of distinct Gaussian primes, Theorem 1: (fundamental theorem of arithmetic) Every integer n ≥ 2 has a factorization as a Product of prime powers:

Number of Ideals and Principal Ideals of a Ring R
Definition: A non-empty subset S of a ring (R, +, *) is called an ideal of R if (S,+) is an abelian group of (R,+) for all s S ∈ then rs & sr S ∈ Definition: An ideal of a ring R is said to be a principal ideal it is generated by single element of R i.e., if a R ∈ then a set generated by "a" or <a> is a principal ideal of r / } { a ax x R R < > = ∀ ∈ ⊆

Number Generators of a Cyclic Group
Proposition: Let G be a cyclic group of order n, then G has ( ) n φ generators.
Theorem: (A product formula for ( ) n φ ) Statement: for 1 n ≥ we have Proof: for n=1 the product is empty since there are no primes which divide 1 In this case it is understood that the product is to be assigned the value 1 ∴ (1) 1 φ = .
Suppose that 1 n > and let , , ........... 1 2 3, p p p p r be distinct prime divisors of n. Now the product can be taken as On the right hand side, in a term such as 1 p p p i j k ∑ it is understood that we consider all possible products p p p i j k of distinct prime factors of n taken three at a time. Also each term on the right hand side of equation (1) is of the form 1 d ± where d is a divisor of n which is either 1 or a product of distinct primes. The numerator 1 ± is exactly ( ) d φ is 6 and (13) φ is 12. This is obvious; gcd of all numbers from 1 to p-1 will be 1 because p is a prime. 2. For two numbers a and b, if gcd (a, b) is 1, then ϕ (ab) = ϕ (a) * ϕ (b). For example ϕ (5) is 4 and ϕ (6) is 2, so ϕ (30) must be 8 as 5 and 6 are relatively prime. 3. For any two prime numbers p and q, ϕ (pq) = (p-1)*(q-1).
This property is used in RSA algorithm. 4. If p is a prime number, then ϕ (p k ) = p k -p k-1 . This can be proved using Euler's product formula.

Conclusions
In number theory, the divisor function is a function that is a sum over the divisor function. It every now and again happens in the investigation of the asymptotic behavior of the Riemann zeta function. The different investigations of the behavior of the divisor function are sometimes called divisor problems.
Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then Φ (mn) = φ (m) φ (n) This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring ℤ/nℤ). It additionally assumes a key job in the meaning of the RSA encryption framework.