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:

\(\displaystyle \frac x y\) \(=\) \(\displaystyle \frac {5 i} {3 + i}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {5 i \paren {3 - 1} } {10}\) Definition of Complex Division
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 + 3 i} 2\) simplifying

$q$ is to be set to one of the Gaussian integers nearest to it.

Thus let $q = i$.

Hence:

\(\displaystyle r\) \(=\) \(\displaystyle x - q y\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 2 i\)


Then:

\(\displaystyle \frac {3 + i} {1 + 2 i}\) \(=\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {3 + 1} \paren {1 + 2 i} } 5\) Definition of Complex Division
\(\displaystyle \) \(=\) \(\displaystyle \frac {5 - 5 i} 5\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle 1 - i\) which is a Gaussian integer

Thus a GCD of $5 i$ and $3 + 1$ is $1 + 2 i$.

From Elements of Euclidean Domain have Greatest Common Divisor, its associates are also GCDs of $5 i$ and $3 + 1$.

Hence the result.

$\blacksquare$


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