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Eisenstein's criterion doesn't apply directly to this polynomial, so I have been looking for substitutions of the form $x+a$, with $a \in \mathbb{Q}$, in order to use Eisenstein's criterion on a 'new' polynomial (If this new polynomial is irreducible over $\mathbb{Q}$, then the original one in question must also be, because a substitution is an automorphism).

Maybe there is a different method entirely. Any help would be appreciated!

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Hint: It is a cubic polynomial, therefore if it wasn't irreducible it must have a root. Now use the rational root test.

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  • $\begingroup$ I need to do this without that theorem $\endgroup$ – Joe Nov 2 '16 at 17:55
  • $\begingroup$ @Joseph: This is an odd restriction, given your statements in the body of the Question that there may be "a different method entirely" and "any help would be appreciated". The rational roots test is most often covered in the second year of high school algebra, while Eisenstein's criterion is usually a topic of abstract algebra at the undergraduate or graduate college level. $\endgroup$ – hardmath Nov 2 '16 at 19:22
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  1. $x^3+x+1$ is an irreducible polynomial over $\mathbb{F}_2$
  2. Your polynomial is irreducible over $\mathbb{F}_2$
  3. Your polynomial is irreducible over $\mathbb{Q}$.
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