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Math Help - field of Integers

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

    What does it mean irreducible over Z_5.?
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