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finite-fields - PerN(z the set of periods Finite elds have...

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Finite fields have cyclic multiplicative groups & NumThy: Primitive Roots : Algebra Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 30 April, 2011 (at 22:27 ) Common notation. Use PoT for “power of two”; the PoTs are 1 , 2 , 4 , . . . . Use N to mean “congruent mod N ”. Let n k mean that n and k are co-prime. Use k •| n for “ k divides n ”. Its negation k r | n means “ k does not divide n .” Use n |• k and n r | k for “ n is/is-not a multiple of k .” Finally, for p a prime and E a natnum: Use double-verticals, p E •|| n , to mean that E is the highest power of p which divides n . Or write n ||• p E to emphasize that this is an assertion about n . Use PoT for Power of Two and PoP for Power of ( a ) Prime . For N a posint, let Φ( N ) mean the set { r [ 1 .. N ] | r N } . The cardinality ϕ ( N ) := | Φ( N ) | is the Euler phi function . (So ϕ ( N ) is the cardinality of the multiplicative group, Φ( N ), in the Z N ring.) Use EFT for the Euler-Fermat Thm , which says: Suppose that integers b N , with N positive. Then b ϕ ( N ) N 1 . Use GCoDiLiCo to mean “Greatest Common Divisor is Linear Combination”: Given a non-void finite set of integers, not all zero, their GCD is some integer linear combination of the given integers. Defn: The order of an element. Suppose S is a semi- group ( with unit ), written multiplicatively, which is not necessarily abelian, nor finite. Written Ord S ( z ) or Ord( z ), the order of an element z in a semigroup S is the infimum of all [ 1 .. ) such that z = 1; this infimum can be . Say that z has period n . if z = 1; thus the order of z is the infimum of all the positive periods of z . Let Per S ( z ) be the set { Z | z = S 1 } Of course, if z has finite order, then z is invertible. Thus a semigroup in which every element has finite order is automatically a group. Consequently, asser- tions which would gain no generality if stated for a semigroup S , are stated for a group G . Integers mod N An integer z can have a multiplicative order mod N IFF z N . Let Ord N ( z ) denote this order, and Per N ( z ) the set of periods. 1 : Prop’n. Suppose posints K •| N and z N . Then Ord K ( z ) •| Ord N ( z ) . Proof. Let k := Ord K ( z ). The GCoDiLiCo implies that Per K ( z ) equals k Z . And z n - 1 |• N |• K . So n k Z . Given a ring-hom h Γ 0 , easily the foward image of the units h ( U ) U 0 , where U, U 0 are the respec- tive units. Some units in U 0 may be missed. E.g, h : Z Z 5 by x 7→ hh x ii 5 . 2 : Prop’n. Fix posints N |• K . Let h : Z N Z K be the surjective ring-hom x 7→ hh x ii K . Then the h -image of mult-group Φ( N ) is all of Φ( K ) . In particular Φ( N ) cyclic = Φ( K ) cyclic . * : Hence, if g is an N -primroot, then hh g ii K is a K - primroot. Proof. Let Q := N K . Take the special case that K Q .
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