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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$\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$ – José Carlos Santos Jun 14 '19 at 6:47
