Show that u2root3 is a un in Zroot3abroot3ab element of ZSol

Show that u=2+root3 is a un in Z[root3]={a+broot3:a,b element of Z}.

Solution

Z 3 R. We will show that Z 3 is a sub-ring of R. We must check that conditions therefor are satisfied. We first verify that Z 3 is closed under addition and multiplication. Let x₁, x₂ Z 3 ; then x₁ = a_i + b_i 3 for some a_i , b_i Z for i = 1, 2. Hence, x₁ + x₂ = (a₁ + a₂) + (b₁ + b₂) 3 Z 3 since a₁ + a₂, b₁ + b₂ Z.

x₁ x₂ = (a₁ b₁ + 2a₂ b₂) + (a₁ b₂ + a₂ b₁) 3 Z 3 since a₁ b₁ + 2a₂ b₂, a₁ b₂ + a₂ b₁ Z. Further, since 0_R = 0 + 03 and 0 Z, we have 0_R Z 3 . Now let x Z 3 be given. Then x = a + b 3 for some a, b Z. Then setting y = a + (b) 3, we have y Z 3 , since a, b Z, and x + y = 0_R. Hence, Z 3 is a sub-ring of R. Then since 2 and 1 Z , 2 +3 is some u_n in Z3.