The set of algebraic numbers is a field. I need help proving  this stunning result.
Let me state it this way.
Let  $\mathbb L$ be a field and $\mathbb K$ be a field with $\mathbb K \subset \mathbb L$
Let $A$ be the set of algebraic numbers ie $A=\{a \in \mathbb L, \exists P \in \mathbb K [X], P(a)=0 \}$
I managed to prove that $A$ is a ring with the following reasoning:
If $a,b \in A$ then $ \mathbb K[a]$ is a ring.
And $ \mathbb K \subset \mathbb K[a] \subset \mathbb K[a][b] $
Some finite-dimension argument(ask details if needed) proves that $\mathbb K[a][b]$ has finite dimension as a vector space over the field $\mathbb K$
Reminding that $\mathbb K[a+b] \subset \mathbb K[a][b]$ and $\mathbb K[ab] \subset \mathbb K[a][b]$ proves that they both have finite dimension, hence $a$ and $b$ algebraic.
How should I proceed with inverses ?

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 $a_nx^n +a_{n-1}x^{n-1}+\cdots +a_1x+a_0=0$. 
