Exercise · ZW-EUCLIDEAN-DOMAIN-006 · ★★☆ Intermediate

In \(\mathbb{Z}[i]\) with \(N(a+bi)=a^2+b^2\), which is a valid division of \(7+3i\) by \(2\) (that is, \(7+3i=2q+r\) with \(r=0\) or \(N(r)<N(2)\))?

Choose one option.

Show answer
  • B. \(q=4+2i,\ r=-1-i\)

Explanation

With \(q=4+2i\) we get \(2q=8+4i\) and \(r=-1-i\), of norm \(2<N(2)=4\). The pairs with \(q=3\) or \(q=2+i\) add up correctly, but their remainders have norm 10; the last does not add up to \(7+3i\).

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Review the concept: Euclidean domain

An integral domain \(D\) with a norm \(N\) (a function from \(D\setminus\{0\}\) to the nonnegative integers) allowing division of \(a\) by \(b\neq0\) with quotient \(q\) and remainder \(r\) zero or of smaller norm than \(b\).

\[a=bq+r,\qquad r=0\ \text{ or }\ N(r)

More practice problems: euclidean domain

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