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lemma for imaginary quadratic fields
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(Theorem)
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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 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 if and only if the integers
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(1) |
are 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 |
- 1
- K. V¨AISÄLÄ: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
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"lemma for imaginary quadratic fields" is owned by pahio.
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Cross-references: numbers, rational numbers, field, odd, integer, negative, unique factorization, ring of integers, imaginary quadratic fields
There is 1 reference to this entry.
This is version 4 of lemma for imaginary quadratic fields, born on 2008-10-27, modified 2008-10-28.
Object id is 11216, canonical name is LemmaForImaginaryQuadraticFields.
Accessed 625 times total.
Classification:
| AMS MSC: | 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers) | | | 11R11 (Number theory :: Algebraic number theory: global fields :: Quadratic extensions) |
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Pending Errata and Addenda
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