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Re: help required -- ring, units, mod... etc
Posted:
Mar 19, 2007 3:40 PM
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Dear Maria,
> > More specifically- which elements are units? Which are not?
> an element e in Z_n is a unit if gcd(e,n)=1, otherwise it is not.
Yep- and:
> > Now- which are square roots of 1?
> 1. the unit must be 1
what about the element p-1?
Any others? Why not? (Look at the group structure of the non-zero
elements).
> Z_1 = {0} => units ={0)
> Z_2 = {0,1} => units ={1)
> Z_3 = {0,1,2} => units ={1,2}
> Z_4 = {0,1,2,3} => units ={1,3}
> Z_5 = {0,1,2,3,4} => units ={1,2,3,4}
> Z_6 = {0,1,2,3,4,5} => units ={1,5}
> Z_7 = {0,1,2,3,4,5,6} => units ={1,2,3,4,5,6}
> Z_8 = {0,1,2,3,4,5,6,7} => units = {1,3,5,7}
> Z_9 = {0,1,2,3,4,5,6,7,8} => units = {1,2,4,5,7,8}
> Z_10 = {0,1,2,3,4,5,6,7,8,9}=> units = {1,3,7,9}
> ........
> from the above, all units in Z_2, Z_3, Z_4, Z_6, Z_8 satisfy [u mod n]^2=1.
> i noticed they all have units which are only prime numbers.
Be careful- perhaps you haven't looked at large enough examples!
(Check Z_16, for example) ...
I have to run-
cheers-
Eric
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