[Math] number of element in a principal ideal domain can be $25/36/35/15$

Question

Could any one tell me number of element in a principal ideal domain can be $25/36/35/15$ ?

I just know a principal ideal domain is generated by a single element. what the knowledge I need to find this result?

Thank you

Answer 1

First, you need to know that any finite integral domain is a field. Then you should know that there is a finite field of cardinality $n$ if and only if $n$ is a prime power.

Note that what you said is probably not what you meant:

a principal ideal domain is generated by a single element.

In any ring $R$, we have that $R$ is equal to the ideal generated by $1_R$, so this is true, but the statement you probably meant is

every ideal of a principal ideal domain is generated by a single element.