Answer to Question #284665 in Abstract Algebra for Bret

Question

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.

Answer 1

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 $\Z_2$ .