Is (x^3−x^2+2x−1) prime in ZZ/(3)[x]?

Vorbeckenuc

Vorbeckenuc

Answered question

2022-07-14

Is ( x 3 x 2 + 2 x 1 ) prime in Z / ( 3 ) [ x ] ?

Answer & Explanation

umgangistbf

umgangistbf

Beginner2022-07-15Added 12 answers

Let F be the field with 3 elements. Since F [ x ] is a unique factorization domain, a principal ideal (f) being prime is equivalent to f being irreducible. In your case f is a cubic polynomial, so being irreducible is equivalent to not having any roots in F. Just check the three values: f ( 0 ) = 1, f ( 1 ) = 1, and f ( 2 ) = 1 (all modulo 3). Thus f is irreducible and the ideal it generates is prime.

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