## Construct elements of infinite multiplicative order in some quadratic integer rings

Show that for the group of units in is infinite by exhibiting an explicit unit of infinite multiplicative order.

We begin with a lemma.

Lemma: If , where is squarefree, , and , then has infinite multiplicative order in . Proof: We prove by induction that if , then and . The base case holds by hypothesis. Now suppose , and that and . Then , and we have and . By induction, has no solution .

Using the lemma, it suffices to find, for each , an element such that and are integers, , , and .

- For , note that .
- For , note that .
- For , note that .
- For , note that .

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## Comments

I think there might be a typo here. In particular, was is d? Should this be D?

That was a typo. Thanks!

Is the norm of a+bw not different in the case when D=5 as it is congruent to 1 mod 4.

When we write the elements as , the norm is is . When is 1 mod 4, the ring of integers is enlarged to include elements of the form where . Now looks different.