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I want to solve that question:

Find a greatest common divisor $d$ of $a = 170 + 32\sqrt{3}$ and $b = 61 + 35\sqrt{3}.$ Then find $f,g \in \mathbb{Z}[\sqrt{3}]$ such that $af + bg = d.$

Note that we defined the norm function as follows: Define the norm function $N:\mathbb{Q}[\sqrt{3}] \rightarrow \mathbb{Q}$ by $N(a_{1} + a_{2}\sqrt{3}) = \operatorname{det} \phi (a_{1} + a_{2}\sqrt{3}) = a_{1}^2 - 3 a_{2}^2.$ And we showed that $N(xy) = N(x)N(y)$ for all $x,y \in \mathbb{Q}[\sqrt{3}].$

My questions are:

a-I found those questions here:

1- Efficiently computing GCDs in $\mathbb{Z}[(1+\sqrt{-19})/2]$ and

2- GCD of two elements in $\mathbb Z \left[\frac{1 + \sqrt{-11}}{2}\right]$

but still I do not understand the general procedure of finding gcd using norm function, could anyone explain that for me please?

b- I know that to solve this part of the question "find $f,g \in \mathbb{Z}[\sqrt{3}]$ such that $af + bg = d.$" in case of $\mathbb{Z}$ we usually take the calculations we did in the first part (using the division algorithm )in reverse direction i.e. from last step to first step, But I do not know how we usually do that in case of $\mathbb{Z}[\sqrt{3}].$ Could anyone clarify that for me please?

My trial so far:

Since $N(a) = 25828 > N(b) = 46 = 2 \times 23$ and since $N(a)$ is not divisible by $23,$ then their common divisor is $2.$ then we will calculate $$\frac{170 + 32 \sqrt{3}}{61 + 35 \sqrt{3}} = \frac{3505}{23} - \frac{1999 \sqrt{3}}{23},$$ But this is not in $\mathbb{Z}[\sqrt{3}].$

1 Answers1

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Here is a partial answer.

Since $d$ divides both $a$ and $b$, $N(d)$ divides both $N(a)$ and $N(b)$ and so $N(d)$ divides $\gcd(N(a),N(b))=2$.

Therefore, $N(d) \in \{ \pm 1, \pm 2 \}$. Since neither $-1$ nor $2$ are norms, $N(d) \in \{ 1, -2 \}$.

Let's try $N(d)=-2=(1+\sqrt3)(1-\sqrt3)$ and so we compute $$ \frac{170+32\sqrt3}{1+\sqrt3} = -37 + 69 \sqrt3, \qquad \frac{61+35\sqrt3}{1+\sqrt3} = 22 + 13 \sqrt3 $$ Thus, $d \sim 1+\sqrt3$ because $2$ does not divide $a$ and $b$.

For the second part, it suffices to solve $ (-37 + 69 \sqrt3)f + (22 + 13 \sqrt3)g=1$ .

lhf
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  • Can you show me the other part of the question please? –  Nov 06 '20 at 11:53
  • Why $2, -1$ are not norms? and why the others are? –  Nov 06 '20 at 11:56
  • $2$ and $-1$ are not norms because $x^2-3y^2 \bmod 3$ is never $2$. But $N(1+\sqrt3)=-2$ and $N(1)=1$. – lhf Nov 06 '20 at 12:04
  • and why specifically we are using $(1 + \sqrt{3})$? –  Nov 06 '20 at 12:06
  • Can you help me in finding a proof for the division property? https://math.stackexchange.com/questions/3895866/show-that-mathbbz-sqrt3-satisfies-the-following-division-property –  Nov 06 '20 at 12:08
  • I need to write this gcd as a linear combination of $a$ and $b$ or should I do that in terms of the norm function? –  Nov 06 '20 at 12:10
  • How can I solve this? I do not know. –  Nov 06 '20 at 12:24
  • And why the norm is never $-1$? –  Nov 06 '20 at 12:32
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    @Confusion $-1 \equiv 2 \bmod 3$ – lhf Nov 06 '20 at 12:38