Find $p(x)\in \mathbb{K}(\alpha)[x]$ of degree 2 such that $p(\beta)=0$

Question

I am trying to do this exercise:

Consider the field extension $F/K$. Let $f(x)\in K[x]$ be a monic and irreducible polynomial of degree 3. If $\alpha, \beta, \gamma$ are three different roots of $f$ in $F$, prove that $\beta$ is a root of a polynomial of degree 2 with coefficients in $\mathbb{K}(\alpha)$.

I really do not know how to approach this problem. Some things I have thought about are:

• For $u\in\{\alpha, \beta, \gamma\}$, we have that Irr(u,K)=f(x).
• |K(u):K|=3, for $u\in\{\alpha, \beta, \gamma\}$, and $\{1,u,u^2\} $is a K-base of $K(u)$.
• If $p(x)\in {K}(\alpha)[x]$ of degree 2 is such that $p(\beta)=0$, then $Irr(\beta,K(\alpha))$ divides $g(x)$.
• Since $f(x)\in K[x]\subset K(\alpha)[x] $ and $f(\beta)=0$,$Irr(\beta,K(\alpha))$ divides $f(x)$.
• Since $f(x)$ is irreducible and of degree 3, none of the roots belong to $K$.

None of these things have taken me anywhere.

Any hints for solving this problem would be very helpful. Thanks in advance.

Answer 1

$\beta $ is a root of $q(x)=(x-\beta)(x-\gamma)$. And $q$ is an element of $K(\alpha)[x]$ as it is the result of the long division of $f$ by $x-\alpha$ which are both elements of $K(\alpha)[x]$.