A question regarding existence of solutions to polynomial equations in fields.

Question

Let $k$ be an odd integer greater than or equal to $3$, and let $F$ be a field. Also, let $m$ be an odd integer such that $k \geq m \geq 3$. If every $k$-th degree polynomial equation in $F$ has a solution in $F$, must every $m$-th degree polynomial equation in $F$ also have a solution in $F$? Or, is there a counterexample, meaning, is there a field $F$ and odd integers $k$ and $m$ with $k \geq m \geq 3$, such that $F$ has solutions for every $k$-th degree polynomials, but there is at least one $m$-th degree equation which has no solutions in $F$?