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More generally the ring of integers of an Algebraic Number Field ''K'', often denoted by O''K'', is the Ring of Algebraic Integer s contained in ''K''. Using this notation, we can write Z = O'''Q''' since '''Z''' as above is the ring of integers of the Field '''Q''' of Rational Number s. And indeed, in Algebraic Number Theory the elements of '''Z''' are often called the "rational integers" because of this. An alternative term is maximal order, since the ring of integers of a number field is indeed the unique maximal Order in the field. The ring of integers O''K'' has an integral basis; by this we mean that there exist ''b''1,...,''b''n ∈ O''K'' (the integral basis) such that each element ''x'' in O''K'' can uniquely be represented as : with ''a''i ∈ Z. EXAMPLES If ζ is a ''p''th Root Of Unity and ''K''=Q(ζ) is the corresponding Cyclotomic Field , then an integral basis of O''K'' is given by (1,ζ,ζ2,...,ζp-1). If ''d'' is a Square-free integer and ''K''=Q(''d''1/2) is the corresponding Quadratic Field , then an integral basis of O''K'' is given by (1,(1+''d''1/2)/2) if d≡1 (mod 4) and by (1,''d''1/2) if ''d''≡2 or 3 (mod 4). |
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