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I tried to follow with SAGE the solvability of solvable quintic polynomial. I took the example of the solvable polynomial $f=T^5+15T+12$, fully studied in Dummit. I understand that the splitting field $L$ is radical with $[L:\mathbb{Q}]=20$.

The square root of the discriminant is (SAGE) $2^53^25^2\sqrt5$ so we have a quadratic subfield $k=\mathbb{Q}(\sqrt{5})\subset L$. This polynomial (as a function) is strictly increasing over $\mathbb{R}$ so has a unique real root $x$. We consider the extension $k(x)/k$ of degree $5$ ($f$ is still irreducible over $k$) so the last extension of the chain of subfields $\mathbb{Q}\subset k=\mathbb{Q}(\sqrt{5})\subset k(x)\subset L$ is quadratic. The polynomial $f$ must split over $k(x)\;(\subset\mathbb{R})$ into a product of the linear factor $(T-x)$ and two quadratic factors $P_1,P_2 \in k(x)[T]$ whose roots are the pair of complex conjugated roots $x_1$,$\bar{x}_1$,$x_2$,$\bar{x}_2$ of $f$.

We can write the euclidean division $f(T)/(T-x)$ in the form: $$P_1P_2=T^4+xT^3+x^2T^2+x^3T+x^4+15$$ $$= \left(T^2 +A(x)T+N_1(x)\right)\;\left(T^2 +C(x)T+N_2(x)\right)\in k(x)[T]$$

How does SAGE gives the coefficients of $P_1=(T^2 +A(x)T+N_1(x))\in k(x)[T]$? The result given by SAGE is: (recall that the ground field is $\mathbb{Q}(\sqrt{5})$ and that $x$ is a root of $T^5+15T+12$, in other words $x^5\equiv -15x-12$)

$A(x)=\frac{\sqrt{5}}{10}x^4 + \frac{\sqrt{5}}{10}x^2 +(\frac{\sqrt{5}}{10} + \frac12) x + \frac{6\sqrt{5}}{5}=x_1+\bar{x}_1$

$N_1(x)=(\frac{\sqrt{5}}{20}+\frac14)x^4+ (\frac{\sqrt{5}}{20} - \frac14) x^3 + (-\frac{3\sqrt{5}}{20} + \frac14)x^2 + (\frac{3\sqrt{5}}{20} - \frac34) x + 3=x_1\bar{x}_1$

and I feal that understanding how SAGE/PARI computes this would give a better understanding of field extensions.

Further, the simplier example for the last quadratic extension $L/k(x)$. It is generated by any other root. Take one root $x_1$, so $x_1^2\in k(x)$, and define the quadratic extension $L=\{k(x)+x_1k(x)\}$ generated by this root. It's conjugate (over $k(x)$) is $x'_1=-A-x_1$. The other pair is $x_2=a+bx_1$ whose conjugate (over $k(x)$) is $x'_2=a'-bx_1$, and both are roots of $P_2=T^2+CT+N_2$ with $x-A=C$ and $xN_1N_2=-12$. How does SAGE finds the values $a,a',b$?

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I though the answer was easier than what I found, so no real chance to do this "by hand". The article from Factoring Polynomials describes a process over $\mathbb{Q}$ called the Zassenhauss method: first factor modulo $p$ (say Berlekamp algorithm, rather accessible), then Hensel lift back and check bounds on coefficients and other tests. The Klueners article describes the "van Hoeij" algorithm which I admit is above my math understanding capabilities.

It also describes the "Trager method" over a field extension. Find a primitive element, compute it's minimal polynomial over $\mathbb{Q}$, factor it with say the previous method, so the extension is the field modulo the maximal ideal generated by the minimal polynomial, and apply chinese remainder to decompose the given polynomial.

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