Prove that the algebraic numbers A = {$x ∈ C|x\text{ is algebraic over }Q$} form a subfield of C.
I'm not sure how to get started on this problem. I know what a subfield is but I don't know how to show that this group is one of C.
Any help would be great, thank you in advance!
There is a theorem is field theory which says that a field extension $E/F$ which is finitely generated by algebraic elements over $F$ is an algebraic extension. From here it is easy to show that algebraic elements are closed under the field operations. If $\alpha,\beta$ are algebraic elements over $F$ (suppose $\beta\ne 0$) then by the theorem $F(\alpha,\beta)/F$ is an algebraic extension. Since $\alpha+\beta,\alpha-\beta,\alpha\beta,\frac{\alpha}{\beta}$ are all elements in $F(\alpha,\beta)$ we conclude they are algebraic over $F$ as well.
So from here we get that the set of algebraic numbers over $\mathbb{Q}$ is a non-empty subset of $\mathbb{C}$ which is closed under all four arithmetic operations. Hence it is a subfield of $\mathbb{C}$.