
A counterexample
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!