Induction - request for help in proving lemma

Question

Could anyone help with proving the following lemma, please?

Let: $n\in \mathbb{N}$, $Z_{n}^{*}:=\{k\in\mathbb{N}: k\in\{1,\dots,n\} \wedge \space GCD(k,n)=1\}$. Then: $\forall n\in \mathbb{N} \space \forall p \in \mathbb{P}: |Z_{p^{n}}^{*}|=p^{n}-p^{n-1}$

I tried to prove this by induction with respect to $n$, but I stuck at general case. I know how induction works, but I can't see how to do main point...

Kenny Wong · Answer 1 · 2017-03-01T17:32:21.563

1

You may be overthinking it. ${\rm gcd \ }(k,p^{n}) = 1$ holds if and only if $k$ is not divisible by $p$.

So you need to count the number of elements in $\mathbb Z_{p^n}$ that are not divisible by $p$.

Ask yourself:

(1) How many elements are there in $\mathbb Z_{p^n}$ in total?

(2) What fraction of these elements are divisible by $p$ and what fraction are not?

edited Mar 01 '17 at 17:32

answered Mar 01 '17 at 17:07

Kenny Wong

33,403

Thanks for help, but I can't stil see how to proceed. If you could give a little bit more of a hint... – Mar 01 '17 at 17:15
i.e. the answer to 2 is ${ p, 2p, 3p, 4p, 5p, \dots p^{n-1}p }$. What fraction of the total number of elements in $\mathbb Z_{p}$ does this represent?? – Kenny Wong Mar 01 '17 at 17:38
The answer to 1) is $p^n$ by the way. I'm not excluding the $p^n$ element (i.e. the zero element). – Kenny Wong Mar 01 '17 at 17:39
Now I see it! This is as follows: – Mar 01 '17 at 17:45
It was just sufficient to consider ring Z_{p}! – Mar 01 '17 at 17:45
So, my solution is: – Mar 01 '17 at 17:46
|Z_{p^n}|=p^{n} (obvious - elementary school level) – Mar 01 '17 at 17:46
Yes, exactly! :) – Kenny Wong Mar 01 '17 at 17:47
set of elements divisble by p :{p,2p,3p,4p,..., p^{p-2}p,p^(p-1)} = A – Mar 01 '17 at 17:47
|A|=p^{n-1}. Ratio of numbers divisible by p to all numbers in Z_{p} is just p^{n-1}/p^n = 1/p. – Mar 01 '17 at 17:49
Finally, all number of elements in Z_{p} not divisible by p is: p^{n} - (1\p)*p^{n-1} = p^{n}-p^{n-1} – Mar 01 '17 at 17:50
How could I have missed that? I did BMO! How come I got stuck? – Mar 01 '17 at 17:51
1

Problem solved. – Mar 01 '17 at 17:52

Induction - request for help in proving lemma

1 Answers1