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December 2nd, 2010, 08:53 PM   #1
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Prime ideal

I'd like an example of a prime ideal of the rationals if there are any.
I thought the only ideal of the rational numbers was the rational numbers.
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December 3rd, 2010, 05:09 AM   #2
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Re: Prime ideal

Hi iamiam,

the rational numbers form a field, thus there are exactly two ideals: itself and the zero ideal: the set containing just zero. The last one is a prime ideal.

That being said, the mathematical language is not perfect: In number theory phrases like "a rational prime ideal" or "a prime ideal in a number field K" typically should be read as "a prime ideal in the ring of integers" respectively "a prime ideal in the ring of algebraic integers in K" sometimes with the trivial prime ideal "0" tacitly excluded. Thus an ideal generated by some prime number p in would often be called a "rational prime ideal" or a "rational prime".

rgds
Peter
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December 5th, 2010, 12:48 PM   #3
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Re: Prime ideal

thanks, that makes perfect sense.
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