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I read that if $F$ is the field of algebraic numbers over $\mathbb{Q}$, then every polynomial in $F[x]$ splits over $F$. That's awesome! Nevertheless, I don't fully understand why it is true. Can you throw some ideas about why this is true?

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    This is the transitivity of algebraic extension. If $L$ is an algebraic extension of $K$ and $K$ is an algebraic extension of $F$ then $L$ is also an algebraic extension of $F$. Here we use $F = \mathbb{Q}$.2016-03-31

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The fact you mention is that the field $\overline{\mathbb{Q}}$ of algebraic numbers is algebraically closed. This is true because it is the algebraic closure of the field $\mathbb{Q}$ inside the larger algebraically closed field $\mathbb{C}$, i.e., the set of all complex numbers which satisfy some nonzero polynomial equation with $\mathbb{Q}$-coefficients.

Notwithstanding the terminology, that the algebraic closure of a field in an algebraically closed field is algebraically closed requires (nontrivial) proof! See for instance $\S 4$ of my notes on field theory, especially Proposition 17 and Corollary 18.

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Consider some polynomial $x^n = \sum_{i=0}^{n-1} c_i x^i,$ where the $c_i$ are algebraic numbers. Thus for each $i$ we have a similar identity $c_i^{n_i} = \sum_{j=0}^{n_i-1} d_{i,j} c_i^j,$ where this time the $d_{i,j}$ are rationals.

Suppose that $\alpha$ is a root of the original polynomial. By using the above identities, every power of $\alpha$ can be expressed as a linear combination with rational coefficients of terms of the form $\alpha^m c_0^{m_0} \cdots c_{n-1}^{m_{n-1}},$ where $0 \leq m < n \text{ and } 0 \leq m_i < n_i.$ Putting all these $N = nn_0\cdots n_{n-1}$ elements into a vector $v$, we get that there are rational vectors $u_k$ such that $\alpha^k = \langle u_k,v \rangle.$ Among the first $N+1$ vectors $u_k$ there must be some non-trivial rational linear combination that vanishes: $\sum_{k=0}^N t_k u_k = 0.$ Therefore $\sum_{k=0}^N t_k \alpha^k = 0,$ and so $\alpha$ is also algebraic.

This proof is taken from these lecture notes, but it's pretty standard.