Answer to Question #284665 in Abstract Algebra for Bret

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

Solution :

Given that ,

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

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

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

Therefore , whenever we write "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)" can't be written as a product of two irreducible polynomials over "\\Z_2" .

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

And "(x+1)^3=x^3+x^2+x+1" over "\\Z_2" .



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