Is there an easy method, given a degree $d$ and discriminant $D$, to find a polynomial $P(x)$ with degree $d$ and discriminant $D$.
The example I am looking to solve is a polynomial $P(x)$, degree $6$ with the discriminant $D=-7^5*29^2*337^2$.
I came across a slightly less harder problem, to find a polynomial $P(x)$, degree $4$ with discriminant $D=5^3*11^2*61^2$.
Please help explain how to or find these polynomials (in general given a specific degree and discriminant.) Thanks.
It's easier than you think: the discriminant of $x^n-c$ is $n^n c^{n-1}$. (One can have lots of fun proving this by grinding out the resultant of $x^n-c$ and its derivative $nx^{n-1}$: it should be sparse enough to make this fairly straightforward, if rather heavy on paper use.)