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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?

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    $\begingroup$ Use Eisenstein's criterion. Also please see math.meta.stackexchange.com/questions/5020/… $\endgroup$
    – Angina Seng
    Aug 23, 2020 at 11:28
  • $\begingroup$ Eisenstein works in its current form. The degree of this polynomial is $7$ which is certianly $>1$. $\endgroup$
    – lulu
    Aug 23, 2020 at 11:37
  • $\begingroup$ 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 $\endgroup$
    – magnus
    Aug 23, 2020 at 11:41
  • $\begingroup$ oh, i understand it now, thanks :) $\endgroup$
    – magnus
    Aug 23, 2020 at 11:43

1 Answer 1

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Apply Eisenstein's Criterion with $p = 2$ to the polynomial $$\color{red}{1} x^7 + \color{blue}{28}x^2 + \color{blue}{32}x + \color{green}6$$

We need to check that

  • $2 \not \mid \color{red}1$
  • $2 \mid \color{blue}{28}$
  • $2 \mid \color{blue}{32}$
  • $2 \mid \color{green}{6}$ and $2^2 \not\mid \color{green}{6}$
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    $\begingroup$ Thanks that helped alot :) $\endgroup$
    – magnus
    Aug 23, 2020 at 12:10

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