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38 1. Fundamentals

Exercises

1. Show that the following are fields with the usual definitions of addi-
   tion and multiplication:

(a) R : the set of all real numbers

(b) Q : the set of all rational numbers

(c) C : the set of all complex numbers x + yi with x, y real, and

i2 = -1.

2. Show that Z is not a field, but is an integral domain.


3. Show that N is not even a ring.


4. Let F[t] denote the set of all polynomials in the variable t whose
   coefficients lie in a field F and for which addition and multiplication
   are defined as in Section 1.1. Thus, Q[t], R[t] and C[t] are the sets
   of polynomials whose coefficients are, respectively, rational, real and
   complex.

(a) Show that Q[t], R[t], C[t] are integral domains.

(b) Show that F[t] is an integral domain.

(c) Interpret Z[t] and show that Z[t] is an integral domain.

(d) Interpret F[ti, t2 ,... , tm]. Show that this is an integral domain.

We say that F[t] is the set of polynomials over F.

5. (a) Show that every field is an integral domain.
   (b) Show that every integral domain satisfies the cancellation law:
   ifuc=bcandc#O,thenu=b.


6. Let m > 2 be a positive integer. The set Z, consists of the numbers
   I&l, 2,3, * > m - 1). We define addition and multiplication on this
   set modulo m:
   u+b=c meansthatO<c<m-landu+bEc(modm)

ub=c meansthatO<c<m-landubEc(modm).

(a) Fill in the addition and multiplication tables for Z7:

+Q123456 -Q123456

0 3 0 0

1 1 4 6 1 4

2 3 0 2 6 3

3 5 2 30 6 5 4

4 4 1 4 4 2 3

5 1 5 4

6 3 6 5