 # Which of the following polynomials in Z_3 [x] is irreducible? (a) p (x) = x ^ 3 + x + 1 (b) p (x) = x ^ 4 + 1 (c) Factorize the polynomials that are not irreducible. Reeves 2021-02-02 Answered
Which of the following polynomials in ${Z}_{3}\left[x\right]$ is irreducible?
(a) $p\left(x\right)={x}^{3}+x+1$
(b) $p\left(x\right)={x}^{4}+1$
(c) Factorize the polynomials that are not irreducible.
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Step 1
Useful result:
If f(x) in $F\left[x\right]$ be a polynomial over a field F of degree two or three.
Then f(x) is irreducible if and only if it has no zeros.
(a) Given $p\left(x\right)={x}^{3}+x+1$
${Z}_{3}=\left\{0,1,2\right\}$
$p\left(0\right)=1,p\left(1\right)=3\text{mod}3=0\text{mod}3=0,p\left(2\right)=11\text{mod}3=2\text{mod}3=2$
Here p(x) has zero over ${Z}_{3}\left[x\right]$.
Therefore, given p(x) is reducible over ${Z}_{3}\left[x\right]$.
Step 2
(b) Given $p\left(x\right)={x}^{4}+1$
This is a reducible polynomial over ${Z}_{3}\left[x\right]$.
(c) Factorization of given polynomials is as follows:
For (a), $p\left(x\right)={x}^{3}+x+1=\left(x+2\right)\left({x}^{2}+x+2\right)\text{mod}3$
For (b), $p\left(x\right)={x}^{4}+1=\left({x}^{2}+x+2\right)\left({x}^{2}+2x+2\right)\text{mod}3$