Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 340 — #348
340
Chapter 9
Number Theory
A sum or product of numbers in ZΠp 5 is in ZΠp 5, and since ZΠp 5 is a
subset of the complex numbers, all the usual rules for addition and multiplication
are true for it. But some weird things do happen. For example, the prime 29 has
factors:
(a) Find x; y 2 ZŒ p 5 such that xy D 29 and x ¤ ˙1 ¤ y.
On the other hand, the number 3 is still a “prime” even in ZŒ p 5. More precisely,
a number p 2 ZΠp 5 is called irreducible over ZΠp 5 iff when xy D p
for some x; y 2 ZŒ p 5, either x D ˙1 or y D ˙1.
Claim. The numbers 3; 2 C p 5, and 2
p
5 are irreducible over ZŒ
p
5.
In particular, this Claim implies that the number 9 factors into irreducibles over
ZΠp 5 in two different ways:
3 3 D 9 D .2 C p 5/.2
p
5/:
So ZΠp 5 is an example of what is called a non-unique factorization domain.
To verify the Claim, we’ll appeal (without proof) to a familiar technical property
of complex numbers given in the following Lemma.
Definition. For a complex number c D r C si where r; s 2 R and i is p 1, the
norm jcj of c is p r 2 C s 2 .
Lemma. For c; d 2 C,
jcdj D jcj jdj :
(b) Prove that jxj 2 ¤ 3 for all x 2 ZŒ p 5.
(c) Prove that if x 2 ZŒ p 5 and jxj D 1, then x D ˙1.
(d) Prove that if jxyj D 3 for some x; y 2 ZŒ p 5, then x D ˙1 or y D ˙1.
Hint: jzj 2 2 N for z 2 ZΠp 5.
(e) Complete the proof of the Claim.
Problems for Section 9.6
Practice Problems
Problem 9.26.
Prove that if a b .mod 14/ and a b .mod 5/, then a b .mod 70/.