reverse Galois?

Question

Which radical expressions of variables a,b,c,..&c. can be written as a polynomial whose coefficients are polynomials in the same variables? For example, I know what to do with $z=a+\sqrt b$ because I can write $(z-a)^2=b$ and then $z^2-2az+a^2-b=0$. But what about $z=a-b\sqrt[3]{d}+\sqrt[5]{c}$, or something much more complicated such as $z=a-b\sqrt[3]{d}+\frac {\sqrt[5]{c}}{f+\sqrt[8]{e-\sqrt[3] g}}$, etc.? Is there an algorithm that will get us the coefficient polynomials of the lowest order polynomial in "z"?

Answer 1

Given a complicated expression built up from simpler ones by field operations, you can build a polynomial over $\mathbb Q(a,b,\ldots)$ that it satisfies. For example, suppose $\alpha$ is a root of polynomial $P(z)$ and $\beta$ is a root of $Q(z)$. Let $A$ and $B$ be the companion matrices of these polynomials (so $\alpha$ and $\beta$ are eigenvalues of $A$ and $B$ respectively). Then $\alpha \beta$ is an eigenvalue of the Kronecker product $A \otimes B$ (and thus a root of its characteristic polynomial), while $\alpha + \beta$ is an eigenvalue of $A \otimes I + I \otimes B$.

The result of this construction might not be irreducible, though. If you factor the polynomial into irreducibles, you can check which irreducible factor has your expression as a root.