We know that a topological space $X$ has a one point compactification if there exist a compact set $Y$ having $X$ as a subspace,where $Y\setminus X$ contains a single point Also for two such $Y$ and $Y'$ we have a homeomorphism which is identity on $X$. Now in the theory of fields,we know for any polynomial $p(x)\in F[x]$ ($F$ is a field) there exist an extension $F'$ of $F$ such that $p$ has root. Also for any two such $F'$ and $F''$ we have an isomorphism between $F'$ and $F''$ which is identity on $F$.
There is too much similarity in the spirit of these two theorem. My question is there any common philosophy behind the proof of these statements?