= Field of algebraic numbers
{title2=$\overline{\mathbb Q}$}
The complex <algebraic numbers> form a <field>, the <algebraic closure> of $\mathbb Q$ inside $\mathbb C$. To see closure under the field operations, if $\alpha,\beta$ are <algebraic numbers>, the <tower law> makes $\mathbb Q(\alpha,\beta)$ a finite-dimensional <field extension> of $\mathbb Q$. Every element of this extension is an <algebraic element>, because its successive powers are <linearly dependent>. In particular it contains $\alpha\pm\beta$, $\alpha\beta$ and $\alpha/\beta$ when $\beta\ne0$.
It is an infinite-degree <algebraic extension>. A primitive $p$th <root of unity> has <minimal polynomial of an algebraic element> $1+X+\cdots+X^{p-1}$ of degree $p-1$ for every prime $p$, so the degrees of its finite subextensions are unbounded. Finally a complex root of a <polynomial> whose coefficients are <algebraic numbers> is itself algebraic over their finite-degree coefficient field and hence over $\mathbb Q$, again by the <tower law>. Thus every such <polynomial> splits in this <field>, establishing the <algebraic closure> assertion.
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