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Ring of Integers


The ring of integers is the set of integers ..., -2, -1, 0, 1, 2, ..., which form a ring. This ring is commonly denoted Z (doublestruck Z), or sometimes I (doublestruck I).

More generally, let K be a number field. Then the ring of integers of K, denoted O_K, is the set of algebraic integers in K, which is a ring of dimension d over Z, where d is the extension degree of K over Q. O_K is also sometimes called the maximal order of K.

The Gaussian integers Z[i]={a+bi:a,b in Z} is the ring of integers of K=Q(i), and the Eisenstein integers Z[omega]={a+bomega:a,b in Z} is the ring of integers of Q(omega), where omega=(-1+sqrt(-3))/2 is a primitive cube root of unity.


See also

Algebraic Integer, Eisenstein Integer, Extension Field Degree, Gaussian Integer, I, Integer, Maximal Order, Number Field, Number Field Order, Ring, Z

Portions of this entry contributed by David Terr

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Cite this as:

Terr, David and Weisstein, Eric W. "Ring of Integers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RingofIntegers.html

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