<s>[INST] Is the polynomial $\frac{1}{64}x^6 + 3x^4 - \frac{1}{4}x^3 - x + 6$ irreducible over $\Bbb Q$? How do I approach proving irreducibility/reducibility in a polynomial with rational coefficients? Can I apply Eisenstein in some way? 
Is the polynomial $\frac{1}{64}x^6 + 3x^4 - \frac{1}{4}x^3 - x + 6$ irreducible over $\Bbb Q$?
 [/INST] Replace $x$ by $2y$ gives
$$\frac{1}{64}x^6 + 3x^4 - \frac{1}{4}x^3 - x + 6 = y^6+48y^4-2y^3-2y+6$$
Now apply Eisenstein criterion with $p=2$.
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