The fact is studied that the ideal class numbers h of types of real quadratic fields usually contain a fixed prime number p as a factor, and the reason is found to be existing there a kind of prime ideals whose p th p...The fact is studied that the ideal class numbers h of types of real quadratic fields usually contain a fixed prime number p as a factor, and the reason is found to be existing there a kind of prime ideals whose p th powers are principal. A modification of the Cohen Lenstra Heuristics for the probability that in this situation the class number h is actually a multiple of p then is presented: Prob (p|h)=1-(1-p -1 )(1-p -2 ).... This idea is also extended to predict the probability that the class P represented by the above prime ideal is actually of order p : Prob (o(P)=p) =1/p. Both of these predictions agree fairly well with the numerical data.展开更多
文摘The fact is studied that the ideal class numbers h of types of real quadratic fields usually contain a fixed prime number p as a factor, and the reason is found to be existing there a kind of prime ideals whose p th powers are principal. A modification of the Cohen Lenstra Heuristics for the probability that in this situation the class number h is actually a multiple of p then is presented: Prob (p|h)=1-(1-p -1 )(1-p -2 ).... This idea is also extended to predict the probability that the class P represented by the above prime ideal is actually of order p : Prob (o(P)=p) =1/p. Both of these predictions agree fairly well with the numerical data.
