Show that sqrt(2) + 1 has a multiplicative inverse in Z[sqrt(2)].

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- Apr 23rd 2010, 12:17 PMNikoBellicmultiplicative inverse
Show that sqrt(2) + 1 has a multiplicative inverse in Z[sqrt(2)].

- Apr 23rd 2010, 12:38 PMchiph588@
- Apr 23rd 2010, 01:31 PMchiph588@
This is kind of overkill but here goes:

You can also do this using norms. has a norm .

Suppose , then .

So we're looking to solve .

Now since for the last equality.

So as you can see to find a solution, we're left to solve a form of Pell's equation, which is solvable i.e. we can find and . Thus has an inverse.

After Pell's equation is solved though, we'll have an answer in terms of and , so this proof needs some polishing at the end. I won't do this however since I've already given you another solution; this posting was just a thought I had.