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
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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.
answered Aug 18 '13 at 12:08
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2$\begingroup$ " any finite integral domain is a field"... +1 for this hint :) So, principal ideal domain is field in this case and as cardinality has to be a prime power, the possible cardinality should be $25=5^2$ ... Thank you so much :) $\endgroup$ – user87543 Oct 24 '13 at 10:35
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25 elements can have a principal ideal domain because for if n is p^2 then there exist a field of order P^2 moreover since a field is pid then every ideal is principal.
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