虚二次域上的不可分正定整Hermite型

Indecomposable Definite Hermite Forms over Imaginary Quadratic Fields

用格论方法证明了虚二次域 F=Q( mi) ( m≡ 3( mod4 )且 m无平方因子 )上存在任意秩 n判别式d(自然数 )的不可分正定整 Hermite型 ,但有下列例外 :Q( 3i) :n=2 ,d=1 ,2 ,4 ,1 0 ;n=3,d=1 ,2 ,5 ;n=4 ,d=1 ,2 ;n=5 ,d=1 ;n=7,d=1 ;Q( 7i) :n=2 ,d=1 ;Q( 1 1 i) :n=2 ,d=2 ;n=3,d=1 ,不存在相应的不可分正定整 Hermite型 .

Utilizing lattice theory , the following results are obtained: with arbitrary nature number n and d, there exist indecomposable positive integer Hermitian forms over Q(mi)(m≡3 (mod 4)) with rank n and discriminant d. But for several exception cases: Q(3i),n=2,d=1,2,4,10; n=3,d=1,2,5;n=4,d=1,2;n=5,d=1;n=7,d=1; Q(7i): n=2,d=1; Q(11i): n=2,d=2;n=3,d=1, there don′t exist the forms with above property.