By VICTOR SHOUP
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Extra resources for A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1)
One denotes by Z∗n the set of all residue classes that have a multiplicative inverse. It is easy to see that Z∗n is closed under multiplication; indeed, if α, β ∈ Z∗n , then (αβ)−1 = α−1 β −1 . 5 in the language of residue classes. For α ∈ Zn and positive integer k, the expression αk denotes the product α · α · · · · · α, where there are k terms in the product. One may extend this deﬁnition to k = 0, deﬁning α0 to be the multiplicative identity n . If α has a multiplicative inverse, then it is easy to see that for any integer k ≥ 0, αk has a multiplicative inverse as well, namely, (α−1 )k , which we may naturally write as α−k .
Find an integer whose multiplicative order modulo 101 is 100. 16. Suppose α ∈ Z∗n has multiplicative order k. Show that for any m ∈ Z, the multiplicative order of αm is k/ gcd(m, k). 17. Suppose α ∈ Z∗n has multiplicative order k, β ∈ Z∗n has multiplicative order , and gcd(k, ) = 1. Show that αβ has multiplicative order k . Hint: use the previous exercise. 18. Prove that for any prime p, we have (p − 1)! ≡ −1 (mod p). 5, we know that the only elements of Z∗p that act as their own multiplicative inverse are [±1]n ; rearrange the terms in the product β∈Z∗p β so that except for [±1]n , the terms are arranged in pairs, where each pair consists of some β ∈ Z∗p and its multiplicative inverse.
X ¯n )”, where x ¯i is the encoding of a list of integers x1 , . . , xn as “(¯ xi . We can also encode lists of lists, and so on, in the obvious way. All of the mathematical objects we shall wish to compute with can be encoded in this way. For example, to encode an n × n matrix of rational numbers, we may encode each rational number as a pair of integers (the numerator and denominator), each row of the matrix as a list of n encodings of rational numbers, and the matrix as a list of n encodings of rows.
A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1) by VICTOR SHOUP