roots of cubic polynomial

Question

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in $k$".

Questions:

(1) Does this really work for an arbitrary field $k$?

(2) Could someone provide a reference or proof?

Answer 1

Over a splitting field of $h$ we have $$h = \lambda(x - \alpha)(x - \beta)(x - \gamma) = \lambda x^3 - \lambda(\alpha + \beta + \gamma)x^2 + \ldots$$ with $\lambda\in k^\times$. So $\alpha + \beta + \gamma\in k$ and therefore, if two roots of $h$ are in $k$, so is the third.

Answer 2

If you have a field, the polynomial is (up to a constant) the product of linear factors for the roots. Thus you can divide off (x-root) for any known root, decreasing the degree. If you know deg-1 roots, the remaining factor is linear and thus also corresponding to a root in the same field.