Question

If$u\in K$ , prove that$F\left(u^2\right)\subseteq F\left(u\right)$ .

Short answer

It is proved that$F\left(u^2\right)\subseteq F\left(u\right)$ .

Step-by-step solution

Definiton of algebraic function

Let $K$be 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 Fu2⊆Fu

Let$u\in K$ .

$F\subseteq F\left(u\right)$and$u\in F\left(u\right)$ .

This implies$u·u\in F\left(u\right)$ .

This gives$u^2\in F\left(u\right)$ .

$u^2$is an element of$F\left(u^2\right)$ that also belongs to$F\left(u\right)$ .

Therefore,$F\left(u^2\right)\subseteq F\left(u\right)$.