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.

Facts before question:

$\textbf{Fact 1:}$ Let $F(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_1X+a_0\in \mathbb{Z}[X]$, with $a_0\neq 0$.

If $|a_{n-1}|>1+|a_{n-2}| + \cdots +|a_1| + |a_0|$, then $F$ irrreducible in $\mathbb{Z}[X]$.

$\textbf{Fact 2:}$ Let $K$ be a field, $F(X,Y)=a_n(X)Y^n + \cdots + a_1(X)Y + a_0\in K[X,Y]$, with $a_o,\ldots,a_{n-1}\in K[X]$, $a_n \in K$ and $a_0a_n\neq 0$.

If $\deg(a_{n-1})>\max\bigl(\{\deg(a_0), \deg(a_1), \ldots, \deg(a_{n-2})\}\bigr)$, then $F$ is irreducible over $K[X]$.

Facts end here.

$\textbf{Conjecture:}$ Let $F(X) = a_nX^n + a_{n-1}X^{n-1} + \cdots + a_1X+a_0\in \mathbb{Z}[X]$, with $a_0\neq 0$.

If $|a_{n-1}|>|a_n|+|a_{n-2}| + \cdots +|a_1| + |a_0|$, then $F$ irrreducible in $\mathbb{Z}[X]$.

Is the above conjecture true? I know nothing about multivariable polynomials, so I'm just looking for a yes or no answer, and, in case it is false, can you please provide a counterexample?

share|improve this question
    
Comment, down voter? –  Git Gud Mar 16 at 16:40
add comment

1 Answer

up vote 3 down vote accepted

Try this for the polynomial $2X^2 + 5X + 2$.

share|improve this answer
    
$2X^2 + 5X + 2 = (X+2)(2X+1)$. Thanks! –  Git Gud Jan 5 '13 at 11:16
add comment

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.