Thread: field of Integers

Hello,

Could you please direct me to a website where I can learn stuff to solve the following type questions.

2 Let Z_2 ={0,1}denote the field of integers mod 2.
Show that the polynomial x^3+x+1 is irreducible over Z_2.
Hence construct a field with eight elements. (You should write down the addition
and multiplication tables of your field.)
3 Prove that the polynomial f (x) = x^2+x+1 is irreducible over Z_5. Hence construct
a field F with 25 elements which extends Z_5. Does the polynomial g(x) = x^2+2 have
a zero in F? In other words, you need to find all the solutions (if any) to the equation
x^2+2 = 0
in F.

Originally Posted by charikaar

2 Let Z_2 ={0,1}denote the field of integers mod 2.
Show that the polynomial x^3+x+1 is irreducible over Z_2.
Hence construct a field with eight elements. (You should write down the addition
and multiplication tables of your field.)

Note got no zero, thus it is irreducible (why?). That means is a field with elements. Each element has the form , where .

Thanks

What does it mean irreducible over Z_5.?