Working in the polynomial ring \mathbb{Z}_{5} [x], give a complete

druczekq4

druczekq4

Answered question

2021-12-04

Working in the polynomial ring Z5 [x], give a complete factorization of f(x)=x4+x3+x2+x+1 into polynomials that are irreducible ove Z5. Make sure to justify your work with an explanation of how you got to your conclusion.

Answer & Explanation

Julie Mathew

Julie Mathew

Beginner2021-12-05Added 15 answers

Step 1
Given that f(x)=x4+x3+x2+x+1
This can be written as,
f(x)=x3(x+1)+x2+(x+1)
=(x3+1)(x+1)+x2
Step 2
So the function is 1,1,1,1,1 are in Z5
Hence f(x)=x4+x3+x2+x+1 is irreducible over Z5.

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