Question

If$u,v\in K$ and$u+v$ is algebraic over $F$, prove that$u$ is algebraic over$F\left(v\right)$ .

Short answer

It is proved that$u$ is algebraic over$F\left(v\right)$ .

Step-by-step solution

Definition of algebraic field

Let$K$ an extension field of$F$ . Then$K$ is said to be an algebraic extension of$F$ if every element of$K$ is algebraic over$F$ .

Show that u is algebraic over Fv 

The sum and difference of two algebraic elements are algebraic. So, let $f\left(x\right)\in F\left[x\right]$.

$u+v$is algebraic over $F$. Therefore,$f\left(u+v\right)=0$ .

Consider the polynomial $g\left(x\right)=f\left(x+v\right)\in F\left(v\right)\left[x\right]$.

Replacing $x$by $u$we get the following:

$\begin{array}{rcl}g\left(u\right)& =& f\left(u+v\right)\\ & =& 0\end{array}$

This gives$g\left(u\right)=0\in F\left(v\right)\left[x\right]$ .

Therefore, there exists a polynomial in$F\left(v\right)\left[x\right]$ such that$g\left(u\right)=0$ .

Hence, $u$is algebraic over$F\left(v\right)$ .