ALgeBralc numBers 821 algebraic integers are algebraic integers,and if an algebraic integer is a rational number it is an ordinary integer We should note that under the new definitions an algebraic integer can contain ordinary fractions.Thus(-13 +V-115)/2 is an algebraic integer of the second degree because it is a root of x2+13x+71 =0.On the other hand (1-V-5)/2 is an algebraic number of degree 2 but not an algebraic integer because it is a root of 2x2-2x +3=0. Dedekind introduced next the concept of a number field.This is a collection F ofreal or complex numbers such that if and B belong to F then so do a+B,a-B,aB and,if B0,a/B.Every number field contains the rational numbers because if a belongs then so does a/a or I and consequently 1+1,1+2,and so forth.It is not difficult to show that the set of all algebraic numbers forms a field. If one starts with the rational number field and is an algebraic number of degree n then the set formed by combining with itself and the rational numbers under the four operations is also a field of degree n.This field may be described alternatively as the smallest field containing the rational numbers and a.it is also called an extension field of the rational numbers.such a field does not consist of all algebraic numbers and is a specific algebraic number field.The notation R()is now common.Though one might expect that the members of R(0)are the quotients f(0)/g(0)where f()and g(x)are any polynomials with rational coefficients,one can prove that if is of degree n, then any member a of R()can be expressed in the form a=ao0m-1+10m-2+·+an-1 where the a are ordinary rational numbers.Moreover,there exist algebraic integers ,2,.,0 of this field such that all the algebraic integers of the field are of the form A101+A202+.+An0, where the A are ordinary positive and negative integers A ring,a concept introduced by Dedekind,is essentially any collection of numbers such that if a and 8 belong,so do a+B,a-B,and aB.The set of all algebraic integers forms a ring as does the set of all algebraic integers of any specific algebraic number field. The algebraic integer a is said to be divisible by the algebraic integer B if there is an algebraic integer y such that a=By.Ifj is an algebraic integer which divides every other integer of a field of algebraic numbers then j is called a unit in that field.These units,which include +1 and-1,are a generalization of the units +1 and-I of ordinary number theory.The algebraic integer is a prime if it is not zero or a unit and if any factorization of a into By,where B and y belong to the same algebraic number field, implies that 8 or y is a unit in that field