[seqfan] Re: Generators of groups of units of prime order fields
Emmanuel Vantieghem
emmanuelvantieghem at gmail.com
Mon Jun 18 16:22:35 CEST 2012
The property you are mentioning is very well known and can be found in
Gauss's *Disquisitiones* : see L.E. Dickson, History of the theory of
numbers, vol I, Chapter VII, p. 183.
Emmanuel.
2012/6/18 <jens at voss-ahrensburg.de>
>
> Hi there,
>
> let p be an odd prime, and let |Z_p be the field with p elements.
> The non-zero elements of |Z_p form a group under multiplication, and that
> group is cyclic.
>
> Next, we consider the elements of |Z_p generating that group (of which
> there are phi(p-1)) add them all up.
>
> Then this sum is congruent to mu(p-1) mod p where mu is the Moebius
> function.
>
> Can anyone tell me why?
>
> Examples:
>
> p = 3. The multiplicative group of |Z_3 is generated by 2 which adds up to
> 2 == -1 (mod 3), and mu(2) = -1.
>
> p = 11. The multiplicative group of |Z_11 is generated by 2, 6, 7, or 8
> which add up to 23 == 1 (mod 11), and mu(10) = mu(2*5) = 1.
>
> p = 13. The multiplicative group of |Z_13 is generated by 2, 6, 7 or 11
> which add up to 26 == 0 (mod 13), and mu(12) = mu(2*2*3) = 0.
>
> Regards,
> Jens
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