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Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$. I'm looking for a simple way to establish if $e \in F$. If $E/F$ ...

Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$. Write, $f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$ and $g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$, for some $n,m ...