Describe an example of each of the following A field with fi

Describe an example of each of the following. A field with finitely many elements. A polynomial which is irreducible over the complex numbers. An integral domain which is not a field, and which has infinitely many units Two different rings with 16 elements.

Solution

(a)

Consider the field:

Z_5 ie residues mod 5. It has elements

0,1,2,3,4

It is a field with 5 elements. We can check each element has an inverse.

1^1=1

2*3=6=1 mod 5

3*2=1 mod 5

4*4=1 mod 5

Hence each non zero element has an inverse.

(c)

Consider the ring: R[x]

ie ring of polynomials over real numbers. It is an integral domain as let;

fg=0 for f,g in R[x]

For product of two polynomials to be zero either f must be 0 of g must be zero. Hence there are non zero divisors. So R[x] is an integral domain.

There are infinitely many units as each constant polynomial :f(x)=c ,where c is a non zero real number is a unit