Is true that the number of elements in Gal$(F(u)/F)$ is equal the number of distincts roots of its minimal polynomial on $F(u)$?

Question

Suppose we have an extension field $K$ over $F$ and $u$ an algebraic element of $K$. If $p(x)$ is its minimal polynomial in $F[x]$, is true that the order of Gal$(F(u)/F)$ equals the number of distinct roots of $p(x)$ in $F(u)$??

Answer 1

What is true that the Galois group permutes these distinct roots. But if you think of all ways to permute these roots (or really all ways to permute $n$ objects), you will see that the number of possibilities equals the factorial of the number of objects, and so is MUCH MUCH bigger.

Now not always for every polynomial does every permutation of the roots extend to an element of the Galois group, but for some polynomials they all do! This disappointing fact is the reason that there is no abc-formula like formula for roots of polynomials of degree 5, see Wikipedia: https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem.