Finding solutions belonging to given field So I am preparing for a weekly test at my uni and I have a problem like this:
Find the solutions of the equations in fields $Z_7, Z_{11}, Z_{13}$.
And I'm thinking about the approach for solving such things. Never did that before and what I'd like to know is what methods I should apply.
Say i have $5x^2 + 5x + 1$. If I wanted to do it the "regular" way, I'd do $\Delta = 25(\mod7) - 20(\mod7) = 4 - 6 = 5$. The problem arises later, when I'd have to use $\frac{-b-\sqrt{\Delta}}{2a}$ for example - how do I take a square root then? And is it even a good path I'm treading?

Since $\mathbb{Z}_{i}$ where $i=7,11,13$ are fields so you can use the fact that if $p(x)=f(x).g(x)$ then atleast one of them should be zero.
In cases when $i$ is small the best strategy is to find a root say $a$ thus observing that $(x-a)$ is a factor and then follow long division to reduce the given polynomial into polynomials of smaller degree.
