822 THE THEORY OF NUMBERS 1800-1900 Now let us see to what extent the fundamental theorem of arithmetic holds.In the ring of alalgebraic integers thereareno primes.Let us consider the ring of integers in a specific algebraic number field R(),say the field a+bv-5,where a and b are ordinary rational numbers.In this field unique factorization docs not hold.For example, 21=3.7=(4+V-5(4-V-5=(1+2V-5(1-2W-5: Each of these last four factors is prime in the sense that it cannot be expressed as a product of the form (c +dv-5)(e +fv-5)with c,d,c,and fintegral. On the other hand let us consider the field a+bv6 where a and b are ordinary rational numbers.If one applies the four algebraic operations to these numbers one gets such numbers.If a and b are restricted to integers one gets the algebraic integers (of degree 2)of this domain.In this domain we can takeas an cquivalent definition of unit that the algebraic integerM is a unit if 1/M is also an algebraic integer.Thus 1,-1,5-2v6,and 5+2V6 are units.Every integer is divisible by any one of the units. Further,an algebraic integer of the domain is prime if it is divisible only by itself and the units.now 6=2.3=√6.V6 It would seem as though there is no unique decomposition into primcs.But the factors shown are not primes.In fact 6=2.3=√6.√6=(2+√⑥(-2+V⑥)(3+V⑥(3-V6). Each of the last four factors is a prime in the domain and unique decom position docs hold in this domain. In the ring of integers of a specific algebraic number field factorization of the algebraic integers into primes is always possible but unigue factorization does not generally hold.In fact for domains of the form a +bv-D,where D may have any positive integral value not divisible by a square,the unique factorization theorem is valid only when D =1,2,3,7,11,19,43,67,and 163,at least for D's up to 10Thus the algebraic numbers themselves do not possess the property of unique factorization 4.The Ideals of Dedekind Having generalized the notion of algebraic number,Dedekind now under- took to restore unique factorization in algebraic number fields by a scheme quite different from Kummer's.In place of ideal numbers he introduced 18.H.M.Stark has shown that the above values of D are the only ones ible See hi