> I have answered everyquestion of > chapter zero in the graduate abstract algeb

Question

> I have answered everyquestion of
> chapter zero in the graduate abstract algebra text except
> one...the book is Dummit and Foote 2nd edition...the
> question is section 0.3 number 10, and it states....
>
> Prove that the number of elements in Z/nZ-{0} is thenumber
> given by the
> Euler-Phi (totient) function...
>
> The problem I am having is proving this with just the
> definition of
> Z/nZ-{0}...Now I have already proven that the fundamental
> definition is equivalent to stating the elements ofZ/nZ-{0}
> are precisely those that are relatively prime to n and ifI
> use this fact the proof is trivial, but is it possible to
> prove that the number of elements in Z/nZ-{0} is thetotient
> of n without using the relatively prime proposition?
>
> Without out using the proposition I have established the
> following bounds.
>
> 0 < a <= b < n in this line a and b are elements
> of Z/nZ-{0}...
>
> for ab = 1 mod n, which implies...
>
> ab - 1 = kn....in this line a and b are now integersbut
> still subject to
> the bounds 0 < a <= b < n...
>
> 1 <= k <= ab - 1 < (n-1)^2...
>
> I am not sure where to go from here but I have only given
> the problem a couple of hours of thought....
>
> I guess my main question is am I going in the wrong
> direction...Must one use the relatively prime propositionto
> prove this text question?
>
> thanks in advance and best wishes
> Patrick Rose
>

Explanation / Answer

Dear Patrick, There is a difference between your question and the one statedin Dummit and Foote (I have the 3-rd ed, but it seems the sameplace). They are not asking about Z/nZ-{0} but about themultiplicative group (Z/nZ)+. By proposition 4 on previous page, itis just the set of integers coprime to n, hence the anwser followsby definition of Euler's function. The proposition you try to prove is incorrect if n is notprime: Z/nZ-{0} contains more elements than just the integerscoprime to n. Please explain me the situation as you see it. Regards, Joseph Shtok

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