Substituting $y=1/x$ to apply Eisenstein's Criterion Apparently, the polynomial $1-3x^3$ can be shown to be irreducible in $\mathbb{Q}[x]$ by using the substitution $y=1/x$ and then applying Eisenstein's Criterion. My question is, how can one apply Eisenstein to the function
$$
1-\frac{3}{x^3}
$$
Obviously multiplying by $x^3$ gives an irreducible polynomial but how does that help me?
 A: Hint: Let $k$ be a field and let $f(x) \in k[x]$ be a non-zero polynomial of degree $n$. Define $\sigma(f)(x) := x^n f(1/x).$ Then $\sigma(f)(x) \in k[x].$ Suppose $f$ has non-zero constant coefficient. Then $f$ is irreducible if and only if $\sigma(f)$ is irreducible.
In this case $f(x) = 1 - 3x^3.$ So $\sigma(f)(x) = x^3 - 3.$ By Eisenstein's Criterion, $\sigma(f)(x)$ is irreducible. Hence $f(x)$ is irreducible.
$\bf{EDIT:}$ (proof of the above mentioned result.)
The following things are easy to verify: (a) $\sigma(fg) = \sigma(f) \sigma (g),$ and (b) $\sigma(\sigma(f)) = f,$ (c) $deg(\sigma(f)) = deg f.$ If $f$ is reducible, then $f = gh$ for some non-constant polynomial $f, g \in k[x].$ Then $\sigma(f) = \sigma(gh) = \sigma (g) \sigma(h)$ and both $\sigma(g), \sigma(h)$ are non-constant polynomials in $k[x].$ On the other hand, assume that $\sigma(f)$ is reducible. Then $\sigma(f) = gh,$ for some non-constant polynomials $g, h \in k[x].$ Then $f = \sigma(\sigma(f)) = \sigma(f) \sigma (g)$ and $\sigma(g), \sigma (f)$ are non-constant polynomials in $k[x].$
