Let be a Euclidean Domain. Let be minimal among the norms of nonzero elements of ( exists because every nonempty set of natural numbers has a least element). Prove that every nonzero element of norm is a unit. Deduce that a nonzero element of norm zero (if it exists) is a unit.

We denote the Euclidean norm on by . Choose with and . By the Division Algorithm, we have for some , with . Since is minimal among the norms of nonzero elements in , we have . Thus , so that is a unit.

Suppose now that some nonzero element has norm zero. Then zero is necessarily minimal among the norms of nonzero elements in , and hence any nonzero element of norm zero is a unit.

## Comments

If you begin by choosing $a = 1$ the norm condition still holds and the proof takes half as long.

Good point.

Do you have ideas for the next exercise which is exercise number 4 in section 8.1 in the text book.

Try this.

Thanks very much. Could you give me some hints for number 10 proving that the quotient ring Z[i]/I is finite for any nonzero ideal I of Z[i].

That one is actually in the pipeline and will be posted soon, but I can give a hint.

Note that since is a Euclidean domain, the ideal must be principal. Choose a generator for . Now show that each coset has a representative whose norm is less than the norm of (using the Euclidean algorithm). Finally, argue that the number of elements of which have norm less than some constant is bounded.

Thanks for reading!

Thanks for your hints, and hope to read your proof soon.

Honestly, thank you very much for your proofs. I have learned a lot from you.