Minimal polynomials of real algebraic multiples of roots of unity.

Question

For roots of unity, the minimal polynomial is given by cyclotomic polynomials. Can we extend this to real algebraic multiples of roots of unity? For $r\omega_p$ I think there must be some way to combine the minimal polynomial of $r$ and minimal polynomial of $\omega_p$ to get the minimal polynomial of $r\omega_p$?

The conjugates of $r\omega_p$ will be the subset of $\{r'\omega : r'\ \text{is a conjugate of}\ r\ \text{and}\ \omega^p = 1\}$. But I could not find a counterexample or proof of whether it is exactly this set or not. We can get a polynomial with this set as roots by Resultant. But how do we check whether it is minimal or not? As @ancientmathematician and @Jyrki show this is not always the minimal set. How to find the minimal set?

Additional question: This proves that degree of $r\omega_p \leq$ degree of $r*\phi(p)$. Can we get an inequality for the other way? i.e. Can the degree of $r\omega_p$ be bounded below as a function of the degree of $r$?

Answer 1

If I understood it correctly you are asking whether the number of conjugates of $r\omega_p$ is always equal to $\phi(p)$ times the number of conjugates of $r$.

This is not necessarily the case. A simple example of that phenomenon is $r=\sqrt5$, $p=5$. This is because here $r\in K=\Bbb{Q}(\omega_5)$. More precisely, with $\omega_5=\cos(2\pi/5)+i\sin(2\pi/5)$ we have the well-known $$ \sqrt5=2(\omega_5+\omega_5^4)+1. $$ As $K$ is cyclic of degree four, the number $r\omega_5=\sqrt5\omega_5\in K$ has at most four conjugates. That is less than the predicted $2\cdot4=8$.

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OTOH we have the following relatively general case, where we do get the predicted number of conjugates. Assume that the normal closure $L$ of $\Bbb{Q}(r)$ and the cyclotomic field $K_n=\Bbb{Q}(\omega_n)$ are linearly disjoint. Basically this means that if $L$ is the splitting field of the minimal polynomial of $r$, then we require that $L\cap K_n=\Bbb{Q}$ (two Galois extensions are linearly disjoint iff they intersect trivially).

Then the compositum of fields $F:=K_nL$ is also Galois over $\Bbb{Q}$, and the Galois group is a direct product of $Gal(K_n/\Bbb{Q})\simeq\Bbb{Z}_n^*$ and $Gal(L/\Bbb{Q})$. If $r'$ is any conjugate of $r$, and $\omega_n^k, \gcd(k,n)=1$ is any conjugate of $\omega_n$, then the direct product property implies that $r'\omega_n^k$ is a conjugate of $r\omega_n$.

Answer 2

Not in a straightforward way, since the root of an arbitrary real number will be (with probability one) transcendental. In particular, if $\alpha \in \mathbb{R}$ is transcendental, then we must also have (e.g.) $\sqrt{\alpha}$ transcendental; a reason being, the algebraic numbers are closed under multiplication, so if we had $\sqrt\alpha$ algebraic, then its square, $\alpha$, would be algebraic, too, which is a clear contradiction by assumption.

Answer 3

There need to be more hypotheses, as @Jyrki's clear answer shows.

In fact the easiest counterexample is, I suppose, to take $r=\omega_p$ itself!

However, you also ask how you could tell if the $\mathbb{Q}$ polynomial with roots $r' \omega$ splits. There is an algorithm to factorise any $\mathbb{Z}$-polynomial into irreducibles, so you could just use that. https://en.wikipedia.org/wiki/Factorization_of_polynomials