If α is algebraic over F then F(α) is algebraic over F

Question

As the title explained,if α is algebraic over F,then there is a polynomial p(x) such that p(α)=0 with all coefficient in F.But how can we just use this to show that every linear combination of α with coefficient in F(i.e.F(α)) is a root of some polynomial with coefficient in F?

Answer 1

Hints:

1) As commented, $\;\dim_{\Bbb F}\Bbb F(\alpha)=n<\infty\;$

2) Take $\;\beta\in\Bbb F(\alpha)\;$ , and let $\;B:=\{1,\beta,\beta^2,\ldots,\beta^n\}\subset\Bbb F(\alpha)\;$ . Then $\;B\;$ has $\;n+1\;$ elements so it is linearly dependent over $\;F\;$ ...