Given an irreducible monic polynomial $f(X) \in \mathbb{Q}[X]$ of degree $d$, we can construct extension of $\mathbb{Q}$ as a quotient field $S = \mathbb{Q}[t] / \langle f(t) \rangle $. Considered as polynomial in $S[X]$, $f[X]$ has some roots (at least one).
I wonder, whether there is a systematic way to list roots of $f[X]$ which belong to $S$ represented as elements of $\mathbb{Q}[t] / \langle f(t) \rangle$ (i.e. polynomials over $t$ with coefficients in $\mathbb{Q}$ of degree less than $d$)? Obviously, one of the roots is just $t$ since
$$ f(t) \equiv 0 \mod f(t) $$
so, can I find the other roots (if exist) in $S$ without doing a complete factorization?
Example 1: for $f(X) = X^4 + 1$ and extension field $S = \mathbb{Q}[t] / \langle t^4 + 1 \rangle $, the roots are $\pm t$, $\pm t^3$
Example 2: for $f(X) = X^2 + 2X + 3$ and extension field $S = \mathbb{Q}[t] / \langle t^2 + 2t + 3 \rangle $, the roots are $t$, $- 2 - t$
Example 3 (thanks @dan_fulea): for $f(X) = X^3 - 2$ and extension field $S = \mathbb{Q}[t] / \langle t^3 - 2 \rangle $, there are no other roots than $t$