In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have roots in $K$ (the failure of $K$ to be algebraically closed)? How much can you use $G$ to determine exactly which polynomials in $K[x]$ will have roots in $K$?
For example, the multiplicative group $\mathbb{R}\setminus\{0\}$ has the torsion subgroup $\{1, -1\}$, which is isomorphic to $C_2$, and the only irreducible polynomials in $\mathbb{R}[x]$ have degrees $1$ and $2$. By comparison, the multiplicative group $\mathbb{C}\setminus\{0\}$ has a torsion subgroup consisting of the $n$th roots of unity for all $n$, which is isomorphic to $\mathbb{Q}/\mathbb{Z}$, and $\mathbb{C}$ is algebraically closed. These two properties would seem to be closely related. However, $\mathbb{Q}\setminus\{0\}$ has the same torsion subgroup as $\mathbb{R}\setminus\{0\}$, but $\mathbb{Q}$ has irreducible polynomials of every degree. So what relation is there?
This is an open-ended question, so I'm just looking for some useful information, not necessarily comprehensive treatments. Thanks!