Transcendental numbers (field extension/sum/product)

Question

Let $K \subset L$ be a field extension and $a,b \in L$ transcendental numbers over $K$.

How to show that $a+b$ and $ab$ are not both algebraic over $K$?

I used this:

My idea was to consider $K \subset K(a+b,ab) \subset K(a,b)$

Then $[K(a,b):K]=[K(a,b):K(a+b,ab)][K(a+b,ab):K]$

Since $[K(a,b):K]=\infty$ for $a,b$ transcendental, it follows that $a+b$ and $ab$ are not both algebraic.

I'm not sure if this argumentation is right.

Or how to conclude that $a+b$ and $ab$ are not both algebraic?

Answer 1

If $a+b=\alpha$ with $\alpha$ algebraic over $K$, then $ab=a(\alpha-a)=\beta$ with $\beta$ algebraic over $K$. But then $K(a)$ is a finite extension of $K(\alpha,\beta)$ and hence $a$ is algebraic over $K$, contradiction.