Ring of integers

In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in .[1] An algebraic integer is a root of a monic polynomial with integer coefficients: .[2] This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .

The ring of integers is the simplest possible ring of integers.[lower-alpha 1] Namely, where is the field of rational numbers.[3] And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.

The next simplest example is the ring of Gaussian integers , consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, is a Euclidean domain.

The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.[4]


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