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单词 LemmaForImaginaryQuadraticFields
释义

lemma for imaginary quadratic fields


For determining the imaginary quadratic fieldsMathworldPlanetmath whose ring of integersMathworldPlanetmath has unique factorization, one can use the following

Lemma.  Let d be a negative integer with  d1(mod4),  p the greatest odd irreducible (http://planetmath.org/Irreducible) integer with  p13|d|  and  q=14(1-d).  In the imaginary quadratic field(d), the factorization of integers is unique (http://planetmath.org/Ufd) if and only if the integers

t2-t+q  (t=1, 2,,p+12)(1)

are irreducible (http://planetmath.org/Irreducible) in the field of the rational numbersPlanetmathPlanetmathPlanetmath.

The lemma yields the below table:

qd=1-4qp12(p+1)the numbers (1)
1-3111
2-7112
3-11113
5-19115
11-433211, 13
17-673217, 19
41-1637441, 43, 47, 53

References

  • 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
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更新时间:2025/5/4 6:00:58