$(5)($i$)$ Let $F$ be a field and $f(x)inF[x]$ be an irreducible polynomial in $F[x].$ Let $$ be a root of $f(x)$ in some extension field of $F.$ Show that the number of distinct roots of $f(x)$ in $F()$ divides the degree of $f(x).$

$($ii$)$ Let KsubeL be a field extension such that $|L$:$K|=49$ and char $K=7.$ Suppose that $u^7$inK for all uinL. Show that Gal$(\frac{L}{K})=1.$