We know that , If "\\mathbb{F}" is a finite field , then
"\\frac{ \\mathbb{F}[x]}{<f(x)>}" would be a finite dimensional vector space over a finite field and thus would be a finite set .
Now "\\frac{\\mathbb{F}[x]}{<f(x)>}" is a domain (has no zero divisiors)
"\\iff <f(x)>" is a prime ideal
"\\iff f(x)=0 \\ or \\ f(x)" is a prime elements of the ring (Because we are in a commutative PID)
"\\iff f(x)" is irreducible
(Because in a PID prime elements are same as irreducible)
Now in your case "f(x)=x^8-1" and "\\mathbb{F}=\\Z_2"
Therefore "f(x)" is not irreducible because 1 is a root of "f(x)" .
Now under mod 2 , "x^8-1=x^8+1"
Now "x^8+1=(x^4)^2+2x^4+1^2"
"=(x^4+1)^2"
Again, "x^4+1=(x^2)^2+2x^2+1^2"
"=(x^2+1)^2"
But ,"x^2+1=x^2+2x+1^2"
"=(x+1)^2"
"\\therefore x^8+1=(x+1)^8=(x-1)^8"
........................(1)
A nonzero element "a" of a ring "R" is said to be zero divisors if there exist a nonzero element "b\\in R" such that
either "ab=0 \\ or \\ ba=0" .
In our case , a zero divisor should be divisior of "x^8-1" and not equal to "x^8-1" .
Therefore the zero divisor of
"\\frac{ \\Z_2[x]}{<x^8-1>}" are "(x-1)+<x^8-1>,(x-1)^2+<x^8-1>"
,"(x-1)^3+<x^8-1>,(x-1)^4+<x^8-1>"
,
"(x-1)^5+<x^8-1>,(x-1)^6+<x^8-1>"
,"(x-1)^7+<x^8-1>"
An element "r(x)+<x^8-1>" is nilpotent if and only if there exist a "n\\in \\Z_{>0}" such that "(x^8-1)| r^n(x)" but "x^8-1" does not divide "r(x)."
Therefore the nilpotent element are all those element of "\\frac{\\Z_2[x]}{<x^8-1>}" which have zero divisor as a factor.
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