units in Z[sqrt(2)] and its Norm

[allan]

i want to show that in Z[sqrt(2)] every element of this form (3+2*sqrt(2))^n is unit if n is nonnegative integer ( n>0)?

i know that the Norm of (3+2*sqrt(2)) is (3+2*sqrt(2))*(3-2*sqrt(2)) which is 1.

so the inverse of (3+2*sqrt(2)) is (3-2*sqrt(2))

which implies inverse of (3+2*sqrt(2))^n is (3-2*sqrt(2))^n, and vice-versa.

and [(3+2*sqrt(2))^n]*[(3-2*sqrt(2))^n]=1, however, this is for every integer n , positive or negative. is there any

way to show that (3+2*sqrt(2))^n is a unit for n must be positive integer (n>0).

 

[daon2]

What exactly are you asking? If n is a non-negative integer, then you have shown that \(\displaystyle (3+2\sqrt{2})^n\) is a unit. You cannot show that \(\displaystyle n\) must be positive, since \(\displaystyle (3+2\sqrt{2})^{-1} = 3-2\sqrt{2}\) has multiplicative inverse \(\displaystyle 3+2\sqrt{2} \in \mathbb{Z}[\sqrt{2}]\).