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  $d\equiv 1(mod4)$,  $p$ 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

$t^2-t+q\mathit{\hspace{1em}\hspace{0.25em}}(t=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },{\displaystyle \frac{p+1}{2}})$ (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

Title	lemma for imaginary quadratic fields
Canonical name	LemmaForImaginaryQuadraticFields
Date of creation	2013-03-22 18:31:23
Last modified on	2013-03-22 18:31:23
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Theorem
Classification	msc 11R11
Classification	msc 11R04
Related topic	ListOfAllImaginaryQuadraticPIDs
Related topic	ClassNumbersOfImaginaryQuadraticFields