"Reducing" spliting fields Let $f(X)=X^n+a_{n-1}X^{n-1}+\ldots a_{1}X+a_{0} \in \Bbb Q[X]$, irreducible in $\Bbb Q[X].$ Let $K_{1},K_{2},\ldots,K_{n}$ be the $n$ roots of $f(X)$ in $\mathbb{C}$. Why $\Bbb Q[K_{1},K_{2},\ldots,K_{n}] = \Bbb Q[K_{1},K_{2}, \ldots, K_{n-1}]?$
I think that i need to use the multiplicative property of dimensions of field extensions but i dont know if i'm right. Any help is appreciated!

The sum of the roots $K_1+\dots+K_{n-1}+K_n$ is $-a_{n-1}\in Q\subseteq Q[K_1,\dots, K_{n-1}]$.
Also  $K_1+\dots+K_{n-1} \in Q[K_1,\dots, K_{n-1}]$.
So $K_n\in Q[K_1,\dots, K_{n-1}]$.
