# Thread: field of Integers

1. ## 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.

2. 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 $\displaystyle x^3+x+1$ got no zero, thus it is irreducible (why?). That means $\displaystyle \mathbb{Z}_2[x]/\left< x^3+x+1\right>$ is a field with $\displaystyle 2^3 = 8$ elements. Each element has the form $\displaystyle a+bx + cx^2+\left<x^3+x+1\right>$, where $\displaystyle a,b,c\in \mathbb{Z}_2$.

3. Thanks

What does it mean irreducible over Z_5.?