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Cross-references: class number formula, analytic, size, algebraic number theory, objects, abelian, operation, integer, inverses, identity element, associativity, properties, group, ideals, multiplication, ideal classes, number field
There are 5 references to this entry.
This is version 10 of ideal classes form an abelian group, born on 2002-07-02, modified 2007-08-22.
Object id is 3151, canonical name is IdealClassesFormAnAbelianGroup.
Accessed 4075 times total.
Classification:
| AMS MSC: | 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers) | | | 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants) |
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Pending Errata and Addenda
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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
![$\displaystyle [\mathfrak{a}] \cdot [\mathfrak{b}]=[\mathfrak{a} \mathfrak{b}] $](http://images.planetmath.org:8080/cache/objects/3151/l2h/img5.png "$\displaystyle [\mathfrak{a}] \cdot [\mathfrak{b}]=[\mathfrak{a} \mathfrak{b}] $"){kind=small}
![$ [\mathfrak{a}] \cdot ([\mathfrak{b}] \cdot [\mathfrak{c}])= [\mathfrak{a}] \cd... ...cdot [\mathfrak{c}]= ([\mathfrak{a}] \cdot [\mathfrak{b}]) \cdot [\mathfrak{c}]$](http://images.planetmath.org:8080/cache/objects/3151/l2h/img8.png "$ [\mathfrak{a}] \cdot ([\mathfrak{b}] \cdot [\mathfrak{c}])= [\mathfrak{a}] \cd... ...cdot [\mathfrak{c}]= ([\mathfrak{a}] \cdot [\mathfrak{b}]) \cdot [\mathfrak{c}]$"){kind=small}
![$ [ {\cal{O}}_K ] \cdot [\mathfrak{b}]= [\mathfrak{b}]=[\mathfrak{b}] \cdot [ {\cal{O}}_K ]$](http://images.planetmath.org:8080/cache/objects/3151/l2h/img9.png "$ [ {\cal{O}}_K ] \cdot [\mathfrak{b}]= [\mathfrak{b}]=[\mathfrak{b}] \cdot [ {\cal{O}}_K ]$"){kind=small}
![$ [\mathfrak{b}]$](http://images.planetmath.org:8080/cache/objects/3151/l2h/img10.png "$ [\mathfrak{b}]$"){kind=small}
![$ [\mathfrak{b}] \cdot [\mathfrak{c}] = [(b)]=[ {\cal{O}}_K ]$](http://images.planetmath.org:8080/cache/objects/3151/l2h/img16.png "$ [\mathfrak{b}] \cdot [\mathfrak{c}] = [(b)]=[ {\cal{O}}_K ]$"){kind=small}
![$ [\mathfrak{a}][\mathfrak{b}]=[\mathfrak{a}\mathfrak{b}]= [\mathfrak{b}\mathfrak{a}]=[\mathfrak{b}][\mathfrak{a}]$](http://images.planetmath.org:8080/cache/objects/3151/l2h/img19.png "$ [\mathfrak{a}][\mathfrak{b}]=[\mathfrak{a}\mathfrak{b}]= [\mathfrak{b}\mathfrak{a}]=[\mathfrak{b}][\mathfrak{a}]$"){kind=small}
