Question

Show that there are infinitely many integers $\mathit{k}$ such that

${\mathit{x}}^{\mathbf{9}}\mathbf{+}\mathbf{12}{\mathit{x}}^{\mathbf{5}}\mathbf{-}\mathbf{21}\mathit{x}\mathbf{+}\mathit{k}$is irreducible in $\mathit{\mathbb{Q}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$

Short answer

It is proved there are infinitely many integers $k$such that $x^9+12x^5-21x+k$

is irreducible in$\mathbb{Q}\left[x\right]$

Step-by-step solution

Eisenstein’s criterion

According to Eisenstein’s criterion assume that $f\left(x\right)=a_n{x}^n+\dots +a_{1}x+a_0$be any non constant polynomial having integer coefficients. The polynomial $f\left(x\right)$is irreducible in $\mathbb{Q}\left[x\right]$if there is a prime

$p$such that $p$divides the coefficients $a_0,a_1,\dots ,a_{n-1}$but it does not divide $a_n$ and$p^2$

does not divide$a_0$

Prove the result

Assume that $m\in \mathbb{Z}$beany integer that is relatively prime to 3. Consider $k=3m$.Then, we have

$3\cancel{|}1$, $3|12$, $3|(-21)$, $3|k$and $3^2|k$

By Eisenstein’s criterion, we can say that $x^9+12x^5-21x+k$is irreducible.

As we know that there are infinitely many integers relatively prime to 3, there are infinitely many integers

$k$such that $x^9+12x^5-21x+k$is irreducible in $\mathbb{Q}\left[x\right]$

Hence, it is proved there are infinitely many integers $k$such that$x^9+12x^5-21x+k$ is irreducible in $\mathbb{Q}\left[x\right]$