Q Modulo Z. And so (z=pz) contains q e qe 1 = ˚(q) elements x q of order qe. Xqe 1 1 has qe 1 roots in z=pz;
Ideal Ring Kuosien Integral Domain Sub Integral Domain Ppt Download from slideplayer.info Now suppose p 3 (mod 4). Although entirely standard, we nd the term primitive root to be somewhat archaic. Extensive experiments seem to imply that primitive sequences of order n≥2 over z/(n) are pairwise distinct modulo 2.
It is also a cogenerator in the category of abelian groups.
The above theorem is false if n is replaced by z,q, or r. For example, to compute 16 ⋅ 30 (mod 11) , we can just as well compute 5 ⋅ 8 (mod 11), since 16 ≡ 5 and 30 ≡ 8. (z;+) is cyclic since it is generated by 1, e.g. See the book for the proof, which follows from using fermat's theorem twice to find that a(p −1)(q 1) is congruent to 1 both modulo p and modulo q.
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