I need your help is proving this statement;
Let R be a unique factorization domain and let a be non zero element of R with a=p_1^(α1)p_2^(α2)...P_n^(αn ), where pi are prime elements and αi are positive integers. Show that the number of divisors of a is the product of (1+αi ) where i=1,2,...,n
May be we can proceed by induction, but how can we start.
Thank you in advance
Thaaaaaank you very much for your help. I will show you my solution:
So we use induction on the number of prime numbers appears in ;
If we have only one prime, then and the divisors of are . Hence the number of divisors of is .
Inductive Step: Suppose that contains prime numbers, and the statement is true for . i.e and the number of divisors of is
Our goal is to show that the statement is true for which contain prime numbers.
We have . Hence By inductive hypothesis, the number of divisors of is and we know that the number of divisors of is . So, the number of divisors of is which is
This is my solution. If there is any mistakes or comments please guide me. Thank you again