# 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 $d$ be a negative integer with  $d\equiv 1\pmod{4}$,  $p$ the greatest odd irreducible (http://planetmath.org/Irreducible) integer with  $p\leqq\sqrt{\frac{1}{3}|d|}$  and  $q=\frac{1}{4}(1\!-\!d)$.  In the imaginary quadratic field $\mathbb{Q}(\sqrt{d})$, the factorization of integers is unique (http://planetmath.org/Ufd) if and only if the integers

 $\displaystyle t^{2}\!-\!t\!+\!q\quad\;\left(t=1,\,2,\,\ldots,\,\frac{p\!+\!1}{% 2}\right)$ (1)

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

The lemma yields the below table:

$q$ $d=1-4q$ $p$ $\frac{1}{2}(p\!+\!1)$ the numbers (1)
$1$ $-3$ $1$ $1$ 1
$2$ $-7$ $1$ $1$ 2
$3$ $-11$ $1$ $1$ 3
$5$ $-19$ $1$ $1$ 5
$11$ $-43$ $3$ $2$ 11, 13
$17$ $-67$ $3$ $2$ 17, 19
$41$ $-163$ $7$ $4$ 41, 43, 47, 53

## References

• 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
Title lemma for imaginary quadratic fields LemmaForImaginaryQuadraticFields 2013-03-22 18:31:23 2013-03-22 18:31:23 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 11R11 msc 11R04 ListOfAllImaginaryQuadraticPIDs ClassNumbersOfImaginaryQuadraticFields