Question:

Suppose Q[sqrt(d)] is a Euclidean field , and alpha, beta are two quadratic integers in Q[sqrt(d)] , so that there exist integers gamma and delta in Q[sqrt(d)] so that alpha = gamma*beta + delta, and

|N(delta)| < |N(beta)|. Show by counterexample that gamma and delta are not necessarily unique.

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Also, do you have any idea about the two questions that I posted earlier - "questions related to Q[sqrt(3)] and Q[sqrt(-)]" ?

I see 0 response for them.

If you have any idea, please do teach me. I still can't figure out how to solve these two questions.

Thank you very much!