Many problems of number theory lead to the important question in the arithmetic of algebraic number fields of decomposition of algebraic numbers into prime factors. We shall define a procedure Dec  which returns the decomposition of algebraic numbers into prime factors in a quadratic filed. The problems of factorization are very closely connected with Fermat's (last) theorem. Historically, it was precisely the problem of Fermat's theorem which led Kummer to his fundamental work on the arithmetic of algebraic numbers. 

It is well known the important role of the number h of divisor classes of algebraic number filed play in the arithmetic of the field. Thus one would like to have an explicit formula for the number h in terms of simpler values which depend on the filed. Although this has not been accomplished for arbitrary algebraic number fields, for certain fields of great interest, such as quadratic fields, such formulas as been found. 

Since all divisors are products of prime divisors and the number of prime divisors is infinite, then to compute the number h in a finite number of steps we must use some infinite processes. This is why, in the determination of h, we shall have to consider infinite products, series and other analytic concepts.