Exercise 12.3.1

Proof. Since the degree in $y$ of $f(y)=y^2-4x^3-x$ is 2, it is sufficient to prove that $f$ has no root in $\mathbb{Q}[x]$, or in other words that $\sqrt{4x^3+x}$ is not a polynomial.

This is equivalent to the impossibility of the equality $4x^3+x=p{(x)}^2$, where $p(x)\in \mathbb{Q}[x]$.

If we assume that $4x^3+x=p^2,p\in \mathbb{Q}[x]$, then the irreducible polynomial $x$ divides $p^2$, therefore it divides $p$, thus $x^2$ divides $p^2$, so $x^2\mid 4x^3+x$, $x\mid 4x^2+1$, $x\mid 1$, which is false.

Conclusion: the polynomial $y^2-4x^3-x$ is irreducible when considered as an element of $\mathbb{Q}(x)[y]$. □