Alternative definitions of algebraically closed Let $k$ be a field and $k[x]$ be the polynomial ring with coefficients from $k$.  Prove whether the following are equivalent:


*

*Every nonconstant polynomial in $k[x]$ has a root in $k$.

*Every root of every nonconstant polynomial in $k[x]$ is in $k$.


It is obvious to me that 2 implies 1.  I appreciate any help on this.  Thanks!
 A: Condition 2 can be better stated as every nonconstant polynomial in $k[x]$ factors as a product of degree $1$ factors.
To see that 1 implies 2, note that this is obvious if the polynomial has degree $1$.
Suppose that if a polynomial has degree $n$, then it can be factored as a product of degree $1$ factors and let $f(x)\in k[x]$ have degree $n+1$.  By condition 1 it has a root $r\in k$.
Can you finish without looking at the spoiler?

 Then $f(x)=(x-r)g(x)$ and $g(x)$ has degree $n$.

A: Let $p\in K[x]$ and let's say $deg(p)=n>0$. By assumption it has a root $\alpha_1\in K$. Then $x-\alpha_1|p$. So there is a polynomial $p_1\in K[x]$ of degree $n-1$ such that $p(x)=(x-\alpha_1)p_1(x)$. If $n-1>0$ then $p_1$ is not constant and then it has a root $\alpha_2\in K$. Hence $x-\alpha_2|p_1$ and then we can write $p_1(x)=(x-\alpha_2)p_2(x)$ where $p_2\in K[x]$ has degree $n-2$. Then $p(x)=(x-\alpha_1)(x-\alpha_2)p_2(x)$. Now if $p_2$ is not constant then take a root of $p_2$ in $K$ and continue by induction. At the end you will get the following equality:
$p(x)=c(x-\alpha_1)(x-\alpha_2)...(x-\alpha_n)$ 
Where $c,\alpha_1,\alpha_2,...,\alpha_n\in K$. The roots of $p$ are $\alpha_1,\alpha_2,...,\alpha_n$ and they are all in $K$. 
