# [seqfan] Generators of groups of units of prime order fields

jens at voss-ahrensburg.de jens at voss-ahrensburg.de
Mon Jun 18 13:38:29 CEST 2012

```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
--
----------------------------------------------------
message sent through VSA Webmail v0.96
(c) 2011 October Labs http://www.october-labs.de

VSA v3.6.7 licensed by XenonServer
for any support visit http://www.XenonServer.de
----------------------------------------------------

```