If $f(x)= x^3-tx-1$, where $t$ is an integer. For which value of $t$ is $f(x)$ irreducible over $\mathbb Q[x]$?
As I think was as; in the field of rational polynomials $\mathbb Q[x]$ (i.e., polynomials $f(x)$ with rational coefficients), $f(x)$ is said to be irreducible if there do not exist two nonconstant polynomials $g(x)$ and $h(x)$ in $x$ with rational coefficients such that $f(x)= g(x)\cdot h(x)$
Since $f$ is of degree $3$, if it is a product of non-constant polynomials $g\cdot h$, then one of $g$ and $h$ is of degree $1$, and the other of degree $2$. So $f$ is irreducible if and only if it has no degree $1$ factors, i.e. if and only if it has no roots in $\mathbb Q$.
By rational root theorem, if $f$ has a rational root $p/q$, where $p$ and $q$ are relatively prime, then $p$ divides $1$, and $q$ divides $1$. This shows all possible rational roots of $f$ are $1$ and $-1$. Therefore $f$ is irreducible if and only if both $1$ and $-1$ are not roots, namely, if and only if $t\ne0$ and $t\ne2$.
Hope this helps.