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Let f (x) = x4 + x3 + x2 + x + 1 ∈ Z2[x]. Prove that f(x) is irreducible over Z2[x] or not?

User Ozkolonur
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Answer and Step-by-step explanation:

Let f(x) = x4 + x3 + x2 +x + 1 Є Z2[x]. Prove that f(x) is irreducible over Z2[x] or not?

Proof:-

Let f(x) = x4 + x3 + x2+ x+1 Є Z2[X].

Then f (0) = 1 = f(1), so f(x) has no roots, By Factor theorem, which states that polynomial f(x) has a factor(x-a) if and only if f(a)=0. Hence, f(x) has no linear factor. If f(x) is reducible, it must have factors of degree 2 and degree 3. But f(x) has no degree 2 factors.

We know that only irreducible quadratic in Z2[X] is x2 + x +1. When we divide f(x) by x2 + x +1 we get a remainder of 1, so x2 + x +1 is not a factor of f(x) therefore f(x) is irreducible.

User Virtually Real
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