# Show that the following polynomial is irreducible over $\mathbb Q$

Show that the following polynomial is irreducible over $$\mathbb Q$$:

$$x^7 + 28x^2 + 32x + 6$$

I know how to use the Eisenstein's Criterion method and testing with roots. The problem is that I can't use the methods here

For Eisenstein's Criterion I need a leading polynomial of degree higher than $$1$$

For the root method I need a polynomial of degree $$2$$ or $$3$$.

How can I solve the problem and what is the name of the method I should use?

• Use Eisenstein's criterion. Also please see math.meta.stackexchange.com/questions/5020/… Aug 23, 2020 at 11:28
• Eisenstein works in its current form. The degree of this polynomial is $7$ which is certianly $>1$.
– lulu
Aug 23, 2020 at 11:37
• i have a bit trouble with this condition for example if p = 2: p does not divide a_n. Everything can you divide 1 with Aug 23, 2020 at 11:41
• oh, i understand it now, thanks :) Aug 23, 2020 at 11:43

Apply Eisenstein's Criterion with $$p = 2$$ to the polynomial $$\color{red}{1} x^7 + \color{blue}{28}x^2 + \color{blue}{32}x + \color{green}6$$
• $$2 \not \mid \color{red}1$$
• $$2 \mid \color{blue}{28}$$
• $$2 \mid \color{blue}{32}$$
• $$2 \mid \color{green}{6}$$ and $$2^2 \not\mid \color{green}{6}$$