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.

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