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