Euclidean Domain/Euclidean Algorithm/Examples/5 i and 3 + i in Gaussian Integers
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Examples of Use of Euclidean Algorithm in Euclidean Domain
The GCD of $5 i$ and $3 + 1$ in the ring of Gaussian integers is found to be:
- $\gcd \set {5 i, 3 + 1} = 1 + 2 i$
and its associates $-1 - 2 i$, $-2 + i$ and $2 - i$.
Proof
Let $x = 5 i$ and $y = 3 + i$.
We need to find $q$ and $r$ such that:
- $x = y q + r$
with:
- $\map \nu r < \map \nu y$
where $\map \nu x := \cmod x^2$
Thus we calculate:
\(\ds \frac x y\) | \(=\) | \(\ds \frac {5 i} {3 + i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {5 i \paren {3 - i} } {10}\) | Definition of Complex Division | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 + 3 i} 2\) | simplifying |
$q$ is to be set to one of the Gaussian integers nearest to it.
Thus let $q = i$.
Hence:
\(\ds r\) | \(=\) | \(\ds x - q y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 2 i\) |
Then:
\(\ds \frac {3 + i} {1 + 2 i}\) | \(=\) | \(\ds \frac {\paren {3 + i} \paren {1 - 2 i} } 5\) | Definition of Complex Division | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {5 - 5 i} 5\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 - i\) | which is a Gaussian integer |
Thus a GCD of $5 i$ and $3 + i$ is $1 + 2 i$.
From Euclidean Domain is GCD Domain, its associates are also GCDs of $5 i$ and $3 + i$.
Hence the result.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 28$. Highest Common Factor: Example $56$