Question #284665

Let f(x)=x3 +x2 +x+1 be an element of Z [x] . Write f(x) as a product of 2

irreducible polynomials over Z2.


1
Expert's answer
2022-01-06T06:39:39-0500

Revision :Theorem : Let FF be a field . If f(x)F[x]f(x)\in F[x] and deg(f(x))deg(f(x)) is 2 or 3 , then f(x)f(x) is reducible over FF if and only if f(x)f(x) has a zero in FF .

Solution :

Given that ,

f(x)=x3+x2+x+1Z[x]f(x)=x^3+x^2+x+1 \in \Z[x] .

Clearly , f(x)Z2[x]f(x)\in \Z_2[x] as coefficient of f(x)f(x) are all belong to Z2\Z_2 .

Again f(x)f(x) is a cubic polynomial .

Therefore , whenever we write f(x)f(x) as a product of two polynomials one must be linear .

Again we know that linear polynomials are reducible over the given field.

Hence , f(x)f(x) can't be written as a product of two irreducible polynomials over Z2\Z_2 .

Note: (x+1)3=x3+1+3x2+3x(x+1)^3=x^3+1+3x^2+3x over Z\Z .

And (x+1)3=x3+x2+x+1(x+1)^3=x^3+x^2+x+1 over Z2\Z_2 .



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