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I was reading the theorem about the existence of an integer $t$, the primitive root modulo prime. The proof seemed a bit confusing. I mean the construction part. Why did not they immediately take $t = xy$ instead of $t = x^{m'}y^{m}$? I think $xy$ also satisfies the requirements. Thanks in advance. Here is the link of the proof:

http://www.math.stonybrook.edu/~scott/blair/Proof_Theorem_5.html#B

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    $\begingroup$ I think you can write down a quick sketch here and underline the problem that you want to ask to be more precise. It would help you to get an answer quickly. :) $\endgroup$ – Kumar Jul 8 '19 at 14:18
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    $\begingroup$ That's very difficult to read. I suspect though that $xy$ need not necessarily work. $\endgroup$ – Lord Shark the Unknown Jul 8 '19 at 14:32
  • $\begingroup$ I will make things easier here. So I have an integer $x$ and $d$ is the smallest integer such that $x^{d} \equiv 1$ (mod p). We have another integer $y$ and the smallest integer $e$ such that $y^{e} \equiv 1$ (mod p). Then We want to construct an integer $t$ for which$f = LCM(d, e)$ is the smallest integer such that $t^{f} \equiv 1$ (mod p). That is what I got from there. So what if we take $t = xy$? $x^{k} \equiv 1$ (mod p) if and only if $k$ is a multiple of $d$ and $y^{l} \equiv 1$ (mod p) if and only if $l$ is divisible by $e$. So $LCM(d, e)$ really satisfies the requirements. $\endgroup$ – shota kobakhidze Jul 8 '19 at 14:47
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Generally it is not true that in an abelian group that if $\,x,y\,$ have order $\,j,k\,$ then $xy$ has order $\,{\rm lcm}(j,k),\,$ e.g. consider the case $\,y = x^{-1}.\,$ But it is true that there exists some element of order $\,{\rm lcm}(j,k),\,$ and this is what is proved there (see here for a few other proofs of order lcm-closure)

Remark $ $ Their proof can be simplified. By here: $ $ if $\,x,y\,$ have order $\,d,e\,$ then there are coprime $\,m',m\in \Bbb N\,$ with $\,(d,e)={m'}\,{m},\ (d/m',\,e/m)=1\,$ so $\,x^{\large m'},\, y^{\large m}$ have coprime orders $\,d/m',\, e/m\,$ therefore their product has order $\ (d/m')(e/m) = de/(d,e) = {\rm lcm}(d,e)$.

Unlike many proofs, the linked proof does not require expensive prime factorization. Instead it employs only gcds so it yields an efficient algorithm to compute $\,m',m.$

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  • $\begingroup$ Note: I checked the outline of the cited proof but I did not verify its correctness. $\endgroup$ – Bill Dubuque Jul 8 '19 at 18:52
  • $\begingroup$ Thanks for the response. So, as I understood their proof is complicated because of the formal correctness reasons from the point of view of group theory? However, in this case $xy$ works but such reasoning could fail in some similar situations? $\endgroup$ – shota kobakhidze Jul 8 '19 at 20:43
  • $\begingroup$ @shotakobakhidze Why do you believe that $xy$ works? $\endgroup$ – Bill Dubuque Jul 8 '19 at 20:59
  • $\begingroup$ Yes, you are right. I realized that it is incorrect if in some step I have only left $y$ which is inverse of $x$ and that fails as you mentioned. Then the answer for them is not $LCM(d, e)$ but 1. Nice point. $\endgroup$ – shota kobakhidze Jul 8 '19 at 21:13
  • $\begingroup$ @shota kobakhidze: The author avoids the issue of $y$ being the inverse of $x$ by choosing $y$ so that the order of $y$ doesn't divide $d$. Since the equation $x^d=1$ has at most $d$ solutions, such a choice is always possible (assuming $d < p-1$, we have $d\le{\large{\frac{p-1}{2}}}$). $\endgroup$ – quasi Jul 8 '19 at 21:28
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Let $x\in \{1,...,p-1\}$, and let $d$ be the order of $x$.

If $d=p-1$, then $x$ is a primitive root, and we're done.

Suppose $d < p-1$.

The plan is to find some element of $t\in\{1,...,p-1\}$ whose order exceeds $d$, and then iterate, using $t$ as the new $x$.

As the author argues, there exists $y\in\{1,...,p-1\}$ whose order doesn't divide $d$.

Let $e$ be the order of $y$.

If $e > d$, we can let $t=y$.

Since $e\not\mid d$, we can't have $e=d$.

Suppose $e < d$.

Your claim is that we can let $t=xy$.

Unfortunately, this doesn't always work.

As you correctly observed, since $e\not\mid d$, we get $\text{gcd}(d,e) < e$, hence $$ \text{lcm}(d,e) = \frac{de}{\text{gcd}(d,e)} = d\left(\frac{e}{\text{gcd}(d,e)}\right) > d $$ Let $f$ be the order of $xy$.

Clearly $f{\,|\,}\bigl(\text{lcm}(d,e)\bigr)$.

However, noting Bill Dubuque's post, and correcting my earlier answer, it's not automatic that $f=\text{lcm}(d,e)$.

In fact, we can't even claim $f > d$.

As an example, letting $p=31,x=7,y=23$, we get

  • $x$ has order $d=15$.$\\[4pt]$
  • $y$ has order $e=10$.$\\[4pt]$
  • $xy$ has order $f=6$.

This shows that your idea of using $xy$ for the next iteration doesn't always work.

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  • $\begingroup$ Well, the main idea is to show that $f>d$. Then we can take $t$ instead of $x$ do the same things and got another $t'$ with $f'>f$. finally we will get the degree equal to $ p - 1$ it is the maximal possible value for that. So is it correct in this case? $\endgroup$ – shota kobakhidze Jul 8 '19 at 15:06
  • $\begingroup$ But the point is that for the originally chosen $x,y$, you don't know that $xy$ is a primitive root. $\endgroup$ – quasi Jul 8 '19 at 15:08
  • $\begingroup$ I don't say that $xy$ will be a primitive root. I just say that $xy$ has a higher minimal degree $f$ such that $xy^{f} \equiv 1$ (mod p) than $x$. $\endgroup$ – shota kobakhidze Jul 8 '19 at 15:13
  • $\begingroup$ $LCM(d, e) * GCD(d, e) = de$ we have $GCD(d, e) < e$ , so $LCM(d, e) > d$. If $LCM(d, e) $ is not equal to $p - 1$, I can take $t$ in place of $x$. why can I do it again this for $t$? the same reason why I could do for $x$. I just don't see what can interfere with this. $\endgroup$ – shota kobakhidze Jul 8 '19 at 15:21
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$ – shota kobakhidze Jul 8 '19 at 15:24

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