I want to check whether $f(x)=x^3+2x^2+x-1$ is irreducible over $\mathbb Q, \mathbb R, \mathbb Z_2, \mathbb Z_3$?
Definitely, since $f(x)$ is a polynomial of degree 3, therefore, if it is reducible over any of the fields, there exists a zero of $f(x)$ on that field. But I do not think there is any zero of the polynomial in any of the given fields.
Am I right?
On
Since the degree of the polynomial is odd, you know that it reduces over $\mathbb{R}$. This is because there is a real root of the polynomial. To determine whether the root is rational you can try to apply the Rational Root Theorem.
For roots over $\mathbb{Z}_2$ and $\mathbb{Z}_3$ try to look for roots. This is easily done because these fields are small.
On
HINT: Recall the Mod $p$ Irreducibility Test. I shall state the test for you.
Let $p$ be a prime and suppose that $f(x)\in \mathbb Z[x]$ with $\deg f(x)\ge 1$. Let $\bar f(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ modulo $p$. If $\bar f(x)$ is irreducible over $\mathbb Z_p$ and $\deg\bar f(x)=\deg f(x)$, then $f(x)$ is irreducible over $\mathbb Q$.
Another Hint: There is a theorem that states,
Let $f(x)\in \mathbb Z[x]$. If $f(x)$ is reducible over $\mathbb Q$, then it is reducible over $\mathbb Z$.
Hopefully these hints help!
Well, in $\Bbb R$ it must have a root, as $\displaystyle\lim_{-\infty}f=-\infty$ and $\displaystyle\lim_{+\infty}f=+\infty$, and also, $x=1$ would do it for $\Bbb Z_3$. Else, correct:)