Show that $F(u+c)=F(u) = F(cu)$

Question

Let $K$ be an extension field of the field $F$. If $u \in K$ and $c \in F$, show that $F(u+c)=F(u)=F(cu).$

Intuitively this makes sense to me, but I cannot seem to find a way to write down a structured proof of the statements.

Can anyone please show me how to go about proving this?

Answer 1

1. $F(u) = F(u+c)$

$u \in F(u+c)$ as $u=(u+c)-c$. Similarly $u+c \in F(u)$

2. $F(u) = F(uc)$

$u \in F(uc)$ as $u=(uc)c^{-1}$. Similarly $uc \in F(u)$