Z The Ring Theintegerinterval [0,n) n Z n Z + , i ( Z n ) under - PowerPoint PPT Presentation

Just Remainders Mathematics for Computer Science MIT 6.042J/18.062J i + j ( Z n ) :: = rem(i + j, n) i i j ( Z n ) :: = rem(i i j, n) Z The Ring The integer interval [0, n) n Z n Z + , i ( Z n ) under is called n the ring of integers mod n Albert R Meyer March 11, 2013 Zn.1 Albert R Meyer March 11, 2013 Zn.2 n arithmetic Z versus Z Z n r(k) abbrevs rem(k,n) 3 + 6 = 2 ( Z 7 ) r(i + j) = r(i) + r(j) ( Z n ) 9 i 8 = 6 ( Z 11 ) r(i i j) = r(i) i r(j) ( Z n ) Albert R Meyer March 11, 2013 Zn.4 Albert R Meyer March 11, 2013 Zn.5 1

Rules for Z ≡ (mod n) versus Z n n (i + j) + k = i + (j + k) associativity i ≡ j (mod n) IFF 0 + i = i identity r(i) = r(j) ( Z n ) i + ( − i) = 0 inverse i + j = j + i commutativity Albert R Meyer March 11, 2013 Zn.6 Albert R Meyer March 11, 2013 Zn.7 Z Z Rules for Rules for n n distributivity (i i j) i k = i i (j i k) associativity 1 i i = i identity i i (j + k) = i i j + i i k i i j = j i i commutativity Albert R Meyer March 11, 2013 Zn.8 Albert R Meyer March 11, 2013 Zn.9 2

Z Rules for * :: = elements of Z Z n n n relatively prime to n no cancellation rule * IFF gcd(i, n) = 1 i ∈ Z 3 i 2 = 8 2 ( Z 10 ) i n IFF i is cancellable in Z 3 ≠ 8 ( Z 10 ) n IFF i has an inverse in Z n Albert R Meyer March 11, 2013 Zn.10 Albert R Meyer March 11, 2013 Zn.11 * :: = elements of Z Euler’s Theorem Z n n relatively prime to n k φ (n) = 1 ( Z n ) * φ (n) :: = Z n * for k ∈ Z n Albert R Meyer March 11, 2013 Zn.12 Albert R Meyer March 11, 2013 Zn.13 3

Lemma 1 Lemma 1 kS = S kS = S proof : for S ⊆ Z n s 1 ≠ s 2 IMPLIES ks 1 ≠ ks 2 k ∈ Z * since k is cancellable n Albert R Meyer March 11, 2013 Zn.14 Albert R Meyer March 11, 2013 Zn.15 Lemma 2 Corollary For i, j ∈ Z n , * = k Z * Z n n * IFF i i j * i, j ∈ Z ∈ Z n n * for k ∈ Z n Albert R Meyer March 11, 2013 Zn.16 Albert R Meyer March 11, 2013 Zn.17 4

permuting Z 9 permuting Z 9 φ (9) = 3 2 -3 = 6 * = * = Z 9 1 2 4 5 7 8 Z 9 1 2 4 5 7 8 1 2 4 5 7 8 2 · 2 4 8 1 5 7 Albert R Meyer March 11, 2013 Zn.18 Albert R Meyer March 11, 2013 Zn.19 Corollary permuting Z 9 * = k Z * Z * = n n Z 9 1 2 4 5 7 8 2 · 2 4 8 1 5 7 * for k ∈ Z 7 · 7 5 1 8 4 2 n Albert R Meyer March 11, 2013 Zn.20 Albert R Meyer March 11, 2013 Zn.21 5

Proof of Euler Proof of Euler Π Z * = Π * Π Z * = Π * k Z k Z n n n n = k φ (n) Π Z * n product Albert R Meyer March 11, 2013 Zn.22 Albert R Meyer March 11, 2013 Zn.23 Proof of Euler Proof of Euler Π Z * = 1 = n k φ (n) = k φ (n) Π Z * n Albert R Meyer March 11, 2013 Zn.24 Albert R Meyer March 11, 2013 Zn.25 6

Proof of Euler k φ (n) 1 = QED Albert R Meyer March 11, 2013 Zn.26 7

MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 20 15 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.