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 Z3[x] is irreducible?
(a) p(x)=x3+x+1
(b) p(x)=x4+1
(c) Factorize the polynomials that are not irreducible.
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Expert Answer

BleabyinfibiaG
Answered 2021-02-03 Author has 118 answers
Step 1
Useful result:
If f(x) in F[x] 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(x)=x3+x+1
Z3={0,1,2}
p(0)=1,p(1)=3mod3=0mod3=0,p(2)=11mod3=2mod3=2
Here p(x) has zero over Z3[x].
Therefore, given p(x) is reducible over Z3[x].
Step 2
(b) Given p(x)=x4+1
This is a reducible polynomial over Z3[x].
(c) Factorization of given polynomials is as follows:
For (a), p(x)=x3+x+1=(x+2)(x2+x+2)mod3
For (b), p(x)=x4+1=(x2+x+2)(x2+2x+2)mod3
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