structural:reformatting, stylistic:tone, linguistic:translation
Field

The sum, difference, product, and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic:

For any two algebraic numbers α, β, this follows directly from the fact that the simple extension Q (γ) {\displaystyle \mathbb {Q} (\gamma)}, for γ {\displaystyle \gamma } being either α + β {\displaystyle \alpha +\beta }, α − β {\displaystyle \alpha -\beta }, α β {\displaystyle \alpha \beta } or (for β ≠ 0 {\displaystyle \beta \neq 0}) α / β {\displaystyle \alpha /\beta }, is a linear subspace of the finite- degree field extension Q (α, β) {\displaystyle \mathbb {Q} (\alpha,\beta)}, and therefore has a finite degree itself, from which it follows (as shown above) that γ {\displaystyle \gamma } is algebraic.

An alternative way of showing this is constructively, by using the resultant.

Algebraic numbers thus form a field Q ¯ {\displaystyle {\overline {\mathbb {Q} }}} (sometimes denoted by A {\displaystyle \mathbb {A} }, but that usually denotes the adele ring).