How to find the "relative" defining polynomial of an extension of number fields?

Question

I'm looking at the number field $L$ with label 16.0.3243658447265625.1 in the LMFDB, with defining polynomial $$f= x^{16} - 5 x^{15} + 14 x^{14} - 30 x^{13} + 57 x^{12} - 100 x^{11} + 157 x^{10} - 215 x^{9} + 250 x^{8} - 240 x^{7} + 183 x^{6} - 110 x^{5} + 57 x^{4} - 30 x^{3} + 16 x^{2} - 5 x + 1$$ The LMFDB tells me it has many intermediate fields, such as $K=\mathbb{Q}(\sqrt{-15})$. I want to calculate the relative discriminant (in SageMaths) to study which primes ramify in $L/K$, but to do so, I have to be able to define $L$ as an extension of $K$ by some irreducible polynomial with coefficients in $K$. Is there a better way to do this than just calculating the minimal polynomial of a root of $f$ over $K$?

Answer 1

It's not hard to do it with Sage. Simply factorize the polynomial $f$ over the field $K$:

K.<a> = QuadraticField(-15)
R.<x> = K[]
f = x^16 - 5*x^15 + 14*x^14 - 30*x^13 + 57*x^12 - 100*x^11 + 157*x^10 - 215*x^9 + 250*x^8 - 240*x^7 + 183*x^6 - 110*x^5 + 57*x^4 - 30*x^3 + 16*x^2 - 5*x + 1
print(f.factor())

Output:

(x^8 + (-1/2*a - 5/2)*x^7 + (a + 2)*x^6 + (-3/2*a - 5/2)*x^5 + (5/2*a + 3/2)*x^4 - 2*a*x^3 + (a + 3)*x^2 + (-1/2*a - 5/2)*x + 1) * (x^8 + (1/2*a - 5/2)*x^7 + (-a + 2)*x^6 + (3/2*a - 5/2)*x^5 + (-5/2*a + 3/2)*x^4 + 2*a*x^3 + (-a + 3)*x^2 + (1/2*a - 5/2)*x + 1)

The minimal polynomial of $x$ over $K$ is just any of the two factors (they are conjugate to each other).