Ring of Integers
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المرجع الالكتروني للمعلوماتيه
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www.almerja.com
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24-7-2020
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Ring of Integers
The ring of integers is the set of integers ..., 
, 
, 0, 1, 2, ..., which form a ring. This ring is commonly denoted 
(doublestruck Z), or sometimes 
(doublestruck I).
More generally, let 
be a number field. Then the ring of integers of 
, denoted 
, is the set of algebraic integers in 
, which is a ring of dimension 
over 
, where 
is the extension degree of 
over 
. 
is also sometimes called the maximal order of 
.
The Gaussian integers ![Z[i]=<span style=]()
{a+bi:a,b in Z}" src="https://mathworld.wolfram.com/images/equations/RingofIntegers/Inline16.gif" style="height:15px; width:149px" /> is the ring of integers of 
, and the Eisenstein integers ![Z[omega]=<span style=]()
{a+bomega:a,b in Z}" src="https://mathworld.wolfram.com/images/equations/RingofIntegers/Inline18.gif" style="height:15px; width:161px" /> is the ring of integers of 
, where 
is a primitive cube root of unity.
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