The set of algebraic numbers is a field.
Solution 1:
Hint: Let $r\ne 0$ be a root of the equation $a_0x^n+a_1x^{n-1}+\cdots +a_n=0$. Then $\frac{1}{r}$ is a root of the equation $a_nx^n +a_{n-1}x^{n-1}+\cdots +a_1x+a_0=0$.
Solution 2:
Hints:
For any $\;0\neq a\in A\;$ :
$$a\in\Bbb K[a]=\Bbb K(a)\implies a^{-1}\in \Bbb K(a)\le A\ldots$$