Algebraic extension over field $F$

Question

Suppose I am given a field $F$ and two irreducible polynomials $f,g\in F[X]$ where $f\not =g$. Let $a$ be a root of $f$. We then can create an algebraic field extension $F(a)$ of $F$. Is there a situation where $g$ is reducible in $F(a)[X]$? If so, can someone give me an example of such polynomials? If no, how can I prove it?

Answer 1

Yes, take $f=g$.

A perhaps less trivial situation is if you think of the minimal polynomial $f$ of $\alpha$ over $F$ and take $\beta \in F$ ; then let $g$ be the minimal polynomial of $\alpha + \beta$. Since $\alpha \in F(\alpha) \setminus F$, so does $\alpha + \beta$. Therefore $g$ and $f$ are irreducible over $F$ but $g$ is reducible over $F(\alpha)$ since it admits the root $\alpha+\beta \in F(\alpha)$.

Hope that helps,