The Unit Groups Of The Quotient Rings Of The Real Quadratic Rings

Let Q be the rational number field.For a square-free integer d with d≠0,1,we denote by K=Q(d^1/2)the quadratic field.Let OK be the ring of algebraic integers of K.A quadratic field Q(d^1/2)is called imaginary if d<0,and it is called real if d>0.Then the ring OK is called an imaginary quadratic ring or a real quadratic ring,respectively.Gauss conjectured the only imaginary quadratic fields,which have the unique factorization property arethosewithd=-1、-2、-3、-7、-11、-19、-43、-67、-163.Thisconjecture was proved by H.Stark in 1967.Gauss also conjectured that there are infinitely many real quadr-atic fields having the unique factorization property.But it hasn’t been proved yet.When Q(d^1/2)has the unique factorization property,the primes ξ in Q(d^1/2)have be completely determined.In this paper,we completely determine the unit groups of the quotient rings OK/<ξn>,where fis a prime in Q(d^1/2),and n>0 is an arbitrary rational integer.Throughout this paper,p is a rational prime with the Legendre symbol（d/p）=-1;π is an element in OK satisfying N（π）=±qwith q is a rational prime and the Legendre symbol（d/q）=1.η is an elementin satisfying N（η）=±e and e>0 is a prime factor of d in Z.If 2 |d,we denote by δ ∈ OK satisfies N（δ）=±2.The main contents and conclusions are as follows:Chapter 1 introduces some basic concepts and lemmas of relevant background knowledge,such as the quadratic field,the ring of algebraic integers.Chapter 2 studies the primes,the equivalence classes and the unit groups of the quotient rin-gs OK/<ξn>for d≡1（mod 8）.First we identify the primes ξ in OK,up to associated elemen-ts are given as follows:ξ=p,π,η and δ.According to the primes,we get the equivalence classes and completely determine the unit groups of the quotient rings.Chapter 3 studies the case for d≡2（mod 8）and d≡6（mod 8）,i.e.,d≡2（mod 4）.We determine the primes ξ in OK,up to associated elements,which are given as follows:ξ=p,π,η.The unit groups of OK/<ξn>are isomorphic to the case for d ≡1（mod 8）with ξ=p,π and η,and η is an element in OK satisfies N（η）=±e and e>0 is a prime odd factor of d in Z.So in this chapter we just determine the unit groups of OK/<ηn>for N（η）=±2.Chapter 4 studies the case for d ≡ 3（mod 8）.We determine the primes ξ in OK,up to associated elements,which are given as follows:ξ=p,π,η and δ,with N（δ）=-2.The unit groups of OK/<ξn>are isomorphic to the case for d≡1（mod 8）with ξ=p,π and η.So in this chapter we just determine the unit groups of OK/<ξn>for N（δ）=-2.Chapter 5 studies the case for d ≡ 5（mod 8）.We determine the primes ξ in OK,up to associated elements,which are given as follows:ξ=p,π,η and 2.The unit groups of Ok/<ξu)are isomorphic to the case for d ≡1（mod 8）with ξ=p,π and η.So in this chapter we just determine the unit groups of OK（ξn）for ξ=2.Chapter 6 studies the case for d ≡ 7（mod 8）.We determine the primes ξ in OK,up to associated elements,which are given as follows:ξ=p,π,η and δ,with N（δ）=2.The unit groups of OK/<ξn)are isomorphic to the case for d≡3（mod 8）with ξ=p,π and η.So in this chapter we just determine the unit groups of OK/<δn>for N（δ）=2.