Abstract Algebra Theory and Applications - Computer Science ...

94 CHAPTER 5 COSETS AND LAGRANGE’S THEOREM5.3 Fermat’s and Euler’s TheoremsThe Euler φ-function is the map φ : N → N defined by φ(n) = 1 for n = 1,and, for n > 1, φ(n) is the number of positive integers m with 1 ≤ m < nand gcd(m, n) = 1.From Proposition 2.1, we know that the order of U(n), the group of unitsin Z n , is φ(n). For example, |U(12)| = φ(12) = 4 since the numbers thatare relatively prime to 12 are 1, 5, 7, and 11. For any prime p, φ(p) = p − 1.We state these results in the following theorem.Theorem 5.11 Let U(n) be the group of units in Z n . Then |U(n)| = φ(n).The following theorem is an important result in number theory, due toLeonhard Euler.Theorem 5.12 (Euler’s Theorem) Let a and n be integers such that n >0 and gcd(a, n) = 1. Then a φ(n) ≡ 1 (mod n).Proof. By Theorem 5.11 the order of U(n) is φ(n). Consequently, a φ(n) = 1for all a ∈ U(n); or a φ(n) −1 is divisible by n. Therefore, a φ(n) ≡ 1 (mod n).□If we consider the special case of Euler’s Theorem in which n = p isprime and recall that φ(p) = p − 1, we obtain the following result, due toPierre de Fermat.Theorem 5.13 (Fermat’s Little Theorem) Let p be any prime numberand suppose that p̸ | a. Thena p−1 ≡ 1(mod p).Furthermore, for any integer b, b p ≡ b (mod p).Historical NoteJoseph-Louis Lagrange (1736–1813), born in Turin, Italy, was of French and Italiandescent. His talent for mathematics became apparent at an early age. LeonhardEuler recognized Lagrange’s abilities when Lagrange, who was only 19, communicatedto Euler some work that he had done in the calculus of variations. That yearhe was also named a professor at the Royal Artillery School in Turin. At the ageof 23 he joined the Berlin Academy. Frederick the Great had written to Lagrange