Let f(x)=x^3+x^2+x+1\in Z_2[x]. Write f(x) as a product of irreducible polynomials over Z_2

sanuluy 2021-09-09 Answered
Let \(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{x}^{{2}}+{x}+{1}\in{Z}_{{2}}{\left[{x}\right]}\). Write f(x) as a product of irreducible polynomials over \(\displaystyle{Z}_{{2}}\)

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Expert Answer

Benedict
Answered 2021-09-10 Author has 18297 answers
Given:
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{x}^{{2}}+{x}+{1}\)
The given polynomial can be factorized as a product two irreducible polynomials as,
\(\displaystyle{x}^{{3}}+{x}^{{2}}+{x}+{1}={x}^{{2}}{\left({x}+{1}\right)}+{\left({x}+{1}\right)}\)
\(\displaystyle={\left({x}+{1}\right)}{\left({x}^{{2}}+{1}\right)}\)
Here, the polynomials (x+1) is in irreducible form which cannot be factorized further.
\(\displaystyle{\left({x}^{{2}}+{1}\right)}\) can be factorized over Z as,
\(\displaystyle{x}^{{2}}+{1}={\left({x}+{i}\right)}{\left({x}-{i}\right)}\)
Here, both (x+i) and (x-i) are in irreducible form.
Hence, the given polynomial can be expressed as a product of irreducible polynomial over Z as,
\(\displaystyle{x}^{{3}}+{x}^{{2}}+{x}+{1}={\left({x}+{1}\right)}{\left({x}-{i}\right)}{\left({x}+{i}\right)}\)
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