Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $p$ be a prime, $f$ be a polynomial of $\mathbb{Z}[x]$. Suppose that $f$ is irreducible in $\mathbb{F}_p[x]$.

My question is : Is $f$ irreducible in $\mathbb{Z}[x]$ ?

This question is originally came from the example 4.29 from the notes Field and Galois theory of J.S.Milne, in that he gave an implication of " a polynomial is irreducibile mod 3(still reducible mod 2) then it is irreducible in $\mathbb{Z}$.

Update I have more question after reading the comments and answer below. So, if $f$ is monic, what is the condition for irreducible mod $p$ for some prime $p$ implying irreducibility of $f$ over $\mathbb{Z}[x]$ ? Must it be "irreducible modulo $p$ for all the prime $p$ (not for some) ?

share|improve this question
2  
Not true, as the counterexamples below show. But if the leading coefficient of $f$ is not divisible by $p$, then yes (because then the degrees of any factors are preserved upon reduction modulo $p$). –  Ted Jan 6 '13 at 4:37

3 Answers 3

Certainly not. For example $x+px^2$ is irreducible mod $p$, as it is equivalent to $x$.

share|improve this answer

To elaborate on Ted's comment, if the leading coefficient is not zero in the finite field, then this actually is true. The easiest way to see why is to consider the contrapositive:

$~\bf($irreducible in $\Bbb F_p[x]\implies$ irreducible in $\Bbb Z[x]\bf)\iff ($reducible in $\Bbb Z[x]\implies$ reducible in $\Bbb F_p[x]\bf)$

If $f(x)=a(x)b(x)$ is true in $\Bbb Z[x]$ then it is true in $\Bbb F_q[x]$; the only obstacle to the latter implication then is if one of the $a(x)$ or $b(x)$ is a scalar after reducing modulo $p$, which can't happen if the degree of $f$ is preserved by the modulo reduction.

share|improve this answer
    
So, if a polynomial $f$ is irreducible mod $p$ for some prime $p$, and reducible mod $q$ for some other prime $q$, we can not say about the irreducibility of $f$ over $\mathbb{Z}[x]$, right ? I still have troubles with Milne's example. –  knot Jan 6 '13 at 8:35

Look at $3x^2-x-2$ with $p=3$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.