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How to check if a number field is a PID

asked 2012-06-06 19:28:05 +0200

Alex C gravatar image

updated 2012-06-06 21:12:50 +0200

Given a NumberField, is there a way to check if its ring of integers is a PID? This could be with a built in method, analogous to is_noetherian(), but if there's a more complicated way to do it that's good too.

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answered 2012-07-21 01:18:43 +0200

vdelecroix gravatar image

updated 2012-07-21 01:30:48 +0200

The method class_number gives you the answer, there is no need for further methods

sage: QuadraticField(2).class_number()
sage: QuadraticField(-5).class_number()

For a number field K, the fractional ideals form a group under multiplication. The class number of K is the number of elements in the quotient of this ring by the principal ideals. In particular, the class number is 1 if and only if the ring of integers of K is a PID.

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Asked: 2012-06-06 19:28:05 +0200

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Last updated: Jul 21 '12