Algebraic Numbers

Classes of Ideals.-Two

ideals p and a are called equivalent if there exist principal ideals (r) and (s) such that p(r) = v(s) If also p is equivalent to r, with p(q) = r(1), then a is equivalent to r since o-(sq)= p(r)(q)= (tr) . Hence all ideals equivalent to a given one are equivalent to each other and are said to form a class of ideals.

Evidently any two principal ideals are equivalent. If cr is equivalent to the principal ideal (I), so that a(s) = (r), the number r of the product ci(s) is in (s), whence r is divisible by s and cr=(r/s). Hence the principal class contains all the principal ideals and no others.

For the above set S determined by 0 = 5, we readily specify the classes of ideals. If (a) is any principal ideal, we have a= m(x+y 0) , where x and y are relatively prime integers. We can choose integers r and s such that in which a is properly represented by Since 2 is not such a value of a, p=[2, 1+ 0] is not a principal ideal. Hence the system S was actually enlarged by the intro duction of ideals. In some other systems Z, every ideal is a principal ideal and then the laws of divisibility of whole numbers hold for 2; since they hold for the essentially equivalent set of all ideals. This happens for systems of numbers involving -I -k when k= i, 2, 3, 7, II, 19, 43, 163, but for no further positive values < 1,500,000 of k lacking square factors, and holds for many negative values of k.

For S all ideals equivalent to p are readily shown to be of type (I.) where now a is properly represented by f = Moreover, all ideals are given by (I.). Consider any such ideal and write 1 for the final integer in (I.). Then is of discriminant - al) = and hence is equivalent to one of the two reduced forms or f of discriminant - 20. Thus a is represented by one of them. Hence there are exactly two classes of ideals.

This example illustrates the identity of the problems to find the number of classes of binary quadratic forms of negative discriminant d and the number of classes of ideals for the system of quadratic integers determined by -Id. For a positive d, a similar result holds if in our criterion p(r) = (s) for the equiva lence of ideals p and a, we impose the restriction that the norm of r/ s is positive (narrow equivalence of ideals).

Arithmetics of Algebras.-We

shall now discuss an impor tant generalization of algebraic numbers. A quaternion q=a+bi+cj+dk was called integral by Lipschitz in his Unter suchungen caber die Summen von Quadraten (Bonn, 1886) if and only if a, b, c, d are integers. This natural definition led to serious difficulties and has been discarded. More fortunate was A. Hurwitz in his important book, Zahlentheorie der Quaternionen (1919), who called q integral not only when a, b, c, d are integers, but also when all four are halves of odd integers. Two such integral quaternions have a greatest common right (or left) divisor. The resulting perfect arithmetic of quaternions leads to simple proofs, as well as interpretations, of the main facts about the number of representations of integers as sums of four squares.

Du Pasquier applied Hurwitz's definition unchanged to many further linear algebras and found that they usually do not possess integral numbers.

The latter unfortunate conclusion is avoided by the new defini tion given by Dickson in his Algebras and their Arithmetics (Chicago, 1923), where general theorems were obtained for the first time. In his Algebren and ihre Zahlentheorie (Zurich, 1927), he gave the following simpler, but equivalent, definition: Let A be any rational associative algebra containing 1. Consider a set S of numbers in A each satisfying an algebraic equation with integral coefficients and having i as leading coefficient, and such that the sum, difference and product of any two numbers of S are all numbers of S. Let S contain 1. Then if S is not contained in a larger such set, its numbers are called the integral numbers of algebra A. In case A is an algebraic field, our integral numbers become the classic integral algebraic numbers of A. Again, if A is the algebra of quaternions, our integral numbers become the in tegral quaternions of Hurwitz (see QUATERNIONS).

The applications include a complete theory of the number of (all or only proper) representations by for t= 1, 3, or 7, and by various forms having cross-products; and the complete solUtion in integers x, • • • , u, • • , Ok of Q=.61 62 • • • 6k, where Q is any one of the mentioned quadratic forms. For example, q=a+bi+cj+dk is said to have the co ordinates a, b, c, d and norm Then all integral solutions of . Ok are derived from Perfect Numbers.-A number is called perfect if it is equal to the sum of its divisors smaller than itself. For example, 6 = 1+2+3 is the first perfect number. No odd perfect number is known, but it has not been proved that none exist. It is easily proved that every even perfect number is of the form I). Euclid (Elem., ix. 36) proved that the latter is a perfect number if 2P- I is a prime. Without justification, Mersenne stated in 1644 that the first i 1 values of p for which I is a prime are 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257. It is now known that 67 should be omitted, and 61, 8g, 107 in cluded while the only cases in doubt are 137, 139, 167, 193, 199, 227, 229, Galois Imaginaries.-Under the topic quadratic residues, we found that there is no integer x such that is divisible by 7. Nevertheless, in 183o Galois invented imaginaries x having this property. Their use was later justified by employing the 49 residues ax+b (a, b=o, I, • • • , 6) obtained from all polynomials in x with integral coefficients by suppressing polynomial mul tiples of both and 7. The theory is given completely in Dickson's Linear Groups (Leipzig, 1901). An account in French occurs in Serret's Algebre.