notes_week3 - Week 3 The group of units 3.1 Euler's...

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Week 3 The group of units 3.1 Euler’s ϕ function and Euler’s Theorem We begin with two definitions, which are really just defining notation for things we’ve already seen. Definition 9. We denote by Z m the set of residue classes modulo m . Addition and multiplication on Z m are defined in the usual way: [ a ] + [ b ] = [ a + b ] And [ a ] · [ b ] = [ a · b ] . We know that these operations are well-defined, by Lemma 2.1. Definition 10. We denote by U m Z m the set of residue classes which are units modulo m . We know that U m is a group under multiplication, that is, 1 U m , a, b U m implies ab U m , and a U m implies that ax = 1 has a solution in U m . The Euler ϕ function 1 simple counts the number of units modulo m . Definition 11. The Euler ϕ -function is defined by ϕ ( m ) = # U m , that is, ϕ ( m ) is the number of residue classes of units modulo m . By convention, we set ϕ (1) = 1. Example 13. If p is a prime, then the residue classes modulo p are [0] , [1] , [2] , ..., [ p - 1] . These are all units, except for [0], and so ϕ ( p ) = p - 1. 1 The name of the Greek letter ϕ is pronounced “fee”. 22
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Example 14. The residue classes modulo 6 are [0] , [1] , [2] , [3] , [4] , [5] . Of these, the only ones that are units are [1] and [5], so ϕ (6) = 2. Example 15. Proceeding as in the last example, you can start making a table of values of the function ϕ : m ϕ ( m ) 1 1 2 1 3 2 4 2 5 4 6 2 7 6 8 4 9 6 10 4 11 10 12 4 13 12 14 6 15 8 . . . . . . On of our first problems will be figuring out how to compute ϕ ( m ) eFciently, without resorting to listing all of the residue classes modulo m , and deciding (by computing gcds) which ones are units (there’s nothing wrong with the method, except that it’s much more time consuming than necessary). In order to motivate this problem, and to see why we might want to be able to compute ϕ ( m ), here is a theorem due to Euler. Notice that this theorem is a direct generalization of Theorem 2.6, since ϕ ( p ) = p - 1 if p is prime. Theorem 3.1. Let m > 0 be an integer, and let a be a unit modulo m (i.e., suppose that gcd( a, m ) = 1 ). Then a ϕ ( m ) 1 (mod m ) . Proof sketch. The proof of this is almost exactly the same as the proof of The- orem 2.6. We let b 1 , ..., b ϕ ( m ) be a list of integers representing the ϕ ( m ) di±erent residue classes of units modulo m (so, let b i Z be coprime for each 1 i ϕ ( m ), and suppose that b i ±≡ b j (mod m ), unless i = j ). Then show that the integers ab 1 , .... , ab ϕ ( m ) 23
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are also units, also representing the ϕ ( m ) distinct residue classes of units modulo m . Then, as before, the products are congruent modulo m : b 1 · b 2 · · · b ϕ ( m ) ( ab 1 )( ab 2 ) · · · ( ab ϕ ( m ) ) (mod m ) .
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notes_week3 - Week 3 The group of units 3.1 Euler's...

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