Polynomial $x^4-2x-1$ is Irreducible Over the Field of Rational Numbers $\Q$

Problems and solutions of ring theory in abstract algebra

Problem 234

Show that the polynomial
\[f(x)=x^4-2x-1\] is irreducible over the field of rational numbers $\Q$.

 
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Proof.

We use the fact that $f(x)$ is irreducible over $\Q$ if and only if $f(x+a)$ is irreducible for any $a\in \Q$.
We prove that the polynomial $f(x+1)$ is irreducible.

We have
\begin{align*}
f(x+1)&=(x+1)^4-2(x+1)-1\\
&=(x^4+4x^3+6x^2+4x+1)-2(x+1)-1\\
&=x^4+4x^3+6x^2+2x-2.
\end{align*}
Then the polynomial $f(x+1)$ is monic and all the non-leading coefficients are divisible by the prime number $2$.

Since the constant term is not divisible by $2^2$, Eisenstein’s criterion implies that the polynomial $f(x+1)$ is irreducible over $\Q$.
Therefore by the fact stated above, the polynomial $f(x)$ is also irreducible over $\Q$.


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