lemma for imaginary quadratic fields
For determining the imaginary quadratic fields whose ring of integers
has unique factorization, one can use the following
Lemma. Let be a negative integer with , the greatest odd irreducible (http://planetmath.org/Irreducible) integer with and . In the imaginary quadratic field, the factorization of integers is unique (http://planetmath.org/Ufd) if and only if the integers
(1) |
are irreducible (http://planetmath.org/Irreducible) in the field of the rational numbers.
The lemma yields the below table:
the numbers (1) | ||||
---|---|---|---|---|
1 | ||||
2 | ||||
3 | ||||
5 | ||||
11, 13 | ||||
17, 19 | ||||
41, 43, 47, 53 |
References
- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).