1. ## distinct powers

Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].

2. Originally Posted by NikoBellic
Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].
Are you looking for this to be done group theoretically (as your language and notation seem to suggest) or number theoretically (as the section you posted in seems to suggest)?

3. Originally Posted by NikoBellic
Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].
My Pell's Equation argument in your other post would show there are infinite inverses, but that argument kind of sucks

4. Originally Posted by NikoBellic
Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].
Suppose $(1+\sqrt2)^n=(1+\sqrt2)^k$ where $n>k$.

We then would have $(1+\sqrt2)^{n-k}=1$ which is impossible since $1+\sqrt2>1\implies (1+\sqrt2)^{n-k}>1$

Now observe $(1+\sqrt2)^n\cdot (\sqrt2-1)^n = 1$.