TITLE: Number of real roots of irreducible polynomials that are solvable by radicals
QUESTION [19 upvotes]: Let $n \geq 3$ be a natural number. Define the set $X_n$ as the set of natural numbers that appear as the number of real roots an irreducible polynomial of degree $n$ over $\mathbb{Q}$ which is solvable by radicals can have.
Example: In case $n=p$ is a prime, we have $X_p= \{1,p \}$.
Is $X_n$ or the cardinality $|X_n|$ also known for other values?
It would be interesting to see the beginning of the sequence $a_n=|X_n|$ for small values of $n$, maybe it appears in the oeis.

REPLY [6 votes]: Jensen On the number of real roots of a solvable polynomial
 includes a proof of: 
Loewy’s theorem. Let $K$ be a real number ﬁeld and $f(X)$ an irreducible polynomial in $K[X]$ of odd degree $n$. If $p$ is the smallest prime divisor of $n$ and the Galois group of $f(X)$ over $K$ is solvable, then $r(f) = 1$ or $n$ or satisﬁes the inequalities $p ≤ r(f)≤ n−p +1.$