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PrimitiveRoots - MA 187 PRIMITIVE ROOTS GARSIA 1 Cyclotomic...

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MA 187: PRIMITIVE ROOTS GARSIA may 20, 2009 1 Cyclotomic Polynomials and Primitive Roots 1. Cyclotomic Polynomials We recall that the Moebius function is defined by setting for any integer m μ ( m ) = Ω ( 1 ) k if m is a product of k distinct primes, and 0 otherwise 1 . 1 For instance the following table gives first ten values of μ m μ 1 1 2 1 3 1 4 0 5 1 6 1 7 1 8 0 9 0 10 1 It is customary to define a partial order on the natural numbers by setting d π n if and only if d divides n . Sometimes the symbol “ | ” is used instead of “ π ”. The importance of the Moebius function derives from the following basic result. Theorem 1.1 If { A n } n 1 and { B n } n 1 are two sequences of numbers related by the equations B n = X d π n A d ( n 1 ) , 1 . 2 then A n = X m π n B m μ ( n m ) ( n 1 ) , 1 . 3 and conversely if the relation 1.3 holds then 1.2 must hold as well . Proof If we set B m = P d π m A d in the right hand side of 1.3 it becomes = X m π n ° X d π m A d ¢ μ ( n m ) . Changing order of summation this can be rewritten as X d A d ° X d π m π n μ ( n m ) ¢ 1 . 4 Now it develops that for any two integers d π n we have X d π m π n μ ( n m ) = n 1 if d = n , and 0 otherwise 1 . 5 The reason for this is that if d π m π n then we can write m = dm 0 and n = dn 0 which allows us to cancel the common factor d and rewrite this sum as X m 0 π n 0 μ ( n 0 m 0 ) . 1 . 6
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MA 187: PRIMITIVE ROOTS GARSIA may 20, 2009 2 Clearly, when d = n then n 0 = 1 and this sum reduces to the single term μ (1) = 1. This gives the first case of 1.5. On the other hand when d is strictly less than n , then in 1.6 the sum runs over all divisors of n 0 which only have distinct prime factors. Since those that have a even number of such factors contribute a 1 to the sum and those that have an odd number of such factors contribute a 1 and there is an equal number of each, their contributions do cancel out completely, yielding the second case of 1.5. We can now use 1.5 in 1.4 and see that the sum there reduces to the single term A n , yielding the desired identity 1.3. If an integer n has the factorization n = p α 1 1 p α 2 2 · · · p α k k where p 1 , p 2 , . . . , p k are distinct primes then it is customary to set Φ ( n ) = (1 1 p 1 )(1 1 p 2 ) · · · (1 1 p k ) n 1 . 7 This formula defines the so called Euler Φ -function and it gives the the number of integers m in the interval [1 , n ] that have no factor in common with n . In symbols, we may express this fact by writing Φ ( n ) = # { m [1 , n ] : ( m, n ) = 1 } where as customary the symbol ( m, n ) denotes the greatest common divisor of m and n . It is not di cult to see that 1.7 may also be rewritten in the form Φ ( n ) = X d π n μ ( d ) n d = X d π n μ ( n d ) d . 1 . 8 An example will su ce to convince the reader of the validity of this identity. Let n = 2 3 3 2 5. Note that the definition 1.1 gives that μ ( d ) vanishes for any d that is divisible by the square of a prime.
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