1. A set of numbers F is said to be a eld with respect to addition and multiplication, if with respect to each operation it satises the following properties:

a) F is closed under addition and multiplication, that is, for all a; b 2 F we have a + b and a  b are both in F.

b) commutitivity: a + b = b + a and a  b = b  a

c) associativity: a + (b + c) = (a + b) + c and a  (b  c) = (a  b)  c

d) unique neutral element: there are neutral elements 0 and 1 for addition and multiplication.

e) Unique inverse element for each given element: and element a of F has unique additive inverse and unique multiplicative inverse.

f) ditributivity of multiplication over addition: a  (b + c) = a  b + a  c:

For a given number n > 1 we dene Zn = f0; 1; 2; : : : ; n????1g. Prove that for any prime p, the set Zp with the addition mod p and multiplication mod p, and congruence mod p, is a eld.  (Note: you may need to quote results from our textbook instead of trying to prove the properties yourselves.)

(Comment: there are three famous elds: Rational numbers, Q , the real numbers, R , and complex numbers, C .

A eld is very similar to real numbers as we can perform Algebra on such elds. These elds are innite sets of numbers and they are useful for modeling of innite phenomena. Resualts in page 62 attempt at showing that Q  is a eld; also see pages 132-135; the textbook attempts to establish that certain other collection of a real numbers is also a eld.

When in comes to modeling nite ideas one needs nite elds. It is known that the only nite elds among the Zn 's are the Zp  for some prime p . There are other methods of construting nite elds of order pk  for any powers of primes, the construction of which uses methods similar to methods of chapter 12.)