What are the generators of the three prime ideals $P_1$, $P_2$, and $P_3$ mentioned in the video?

To lift prime ideals from a quotient ring $R/I$ back to the original ring $R$, you take the generators of the prime ideals in $R/I$ and combine them with the generators of the ideal $I$. In this specific problem, the prime ideals in $R$ that contain $I=3R$ are generated by 3 and each of the irreducible factors of the simplified polynomial $\bar{f}$.

Conditions: $R$ is the original ring.; $I = 3R$ is an ideal in $R$.; The irreducible factors of $\bar{f}$ have been found in the quotient ring.

An ideal P of a commutative ring R is prime if it satisfies two properties: first, if the product ab of two elements a and b in R is in P, then at least one of a or b must be in P; second, P is not the entire ring R.

Conditions: R is a commutative ring.; P is an ideal of R.; For all a, b in R, if ab is in P, then a is in P or b is in P.; P is not equal to R.

In the context of a polynomial ring over a field modulo a polynomial, the prime ideals of the resulting quotient ring correspond to the ideals generated by the irreducible factors of the modulus polynomial. Specifically, the prime ideals of $\mathbb{Z}_3[x]/\langle \bar{f} \rangle$ are generated by the irreducible factors of $\bar{f}$.

Conditions: The base ring is a polynomial ring over a field.; The modulus polynomial is factored into irreducibles.