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.
Revision :Theorem : Let be a field . If and is 2 or 3 , then is reducible over if and only if has a zero in .
Solution :
Given that ,
.
Clearly , as coefficient of are all belong to .
Again is a cubic polynomial .
Therefore , whenever we write as a product of two polynomials one must be linear .
Again we know that linear polynomials are reducible over the given field.
Hence , can't be written as a product of two irreducible polynomials over .
Note: over .
And over .
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