Each algebraic integer belongs to the ring of integers of some number field. A number α is an algebraic integer if and only if the ring is finitely generated as an abelian group, which is to say, as a - module. The following are equivalent definitions of an algebraic integer.

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 .

代数整数环(ring of algebraic integers)亦称整数环，是一种特殊的交换整环，代数数域K中的代数整数全体OK称为K的整数环，K是OK的商域，设L⊃K是两个数域，则OL是OK在L的整闭包，OL也是有限生成的OK模，OK是戴德金环，其理想可惟一(不计次序)分解为其素理想的乘积，OK ...

As an important algebraic structure in modern algebra, integer rings are very important in groups, rings, and fields. However, the structure, properties and applications of integer rings await further study. This article mainly gives the construction of integer rings and proves their basic properties. At the same time,

An algebraic integer is a complex number that is the root of monic polynomial with integer coefficients. Show that the set of algebraic integers is a subring of $C$. (Hint: Use symmetric function theorem). I also know that $\alpha \in$ $C$ is an algebraic integer if and only if $m_\alpha,_Q \in Z[x].$ Thanks for the help.

The algebraic integers form an integrally closed ring, meaning that every monic polynomial with coe cients in Z factors down to linear terms over Z, i.e., its roots lie in Z.

Perhaps the simplest example of such a ring is the following: Definition 6.1. The Gaussian integers are the set Z[i] = fx + iy : x,y 2Zgof complex numbers whose real and imaginary parts are both integers. Z[i] is a ring (really a subring of C) since it …

2025年1月31日 · 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 …

The ring of integers O = O(K) is the intersection of the ring of all algebraic integers with K. Note that K = Q[α] for some α where the degree of the minimal polyno-

The Ring of Integers. Elementary number theory is largely about the ring of integers, denoted by the symbol . The integers are an example of an algebraic structure called an integral domain. This means that satisfies the following axioms: (a) has operations + (addition) and (multiplication).