Proof that $z+z^2$ is algebraic over a field

Question

I need to show / or disprove, that if $z\in L$ for a Field extension $L\supseteq K$ is algebraic over $K$, then $z+z^2$ is also algebraic over $K$.

I thought of writing the expression $\sum_{i \in I} k_i(z+z^2)^i$ and forming it around to get a solution I can work with.. but I kinda doubt that this works after trying for a while..

can someone give a hint?

kind regards

Answer 1

Sum and product of algebraic numbers are algebraic, to see this, if $a,b$ are algebraic, it is equivalent to $[K(a):K]$ is finite dimensional, $K(a,b):K(a)]$ is finite dimensional and $[K(a,b):K]\leq [K(a,b):K(a)][K(a):K]$, you deduce that $K(a,b)$ is algebraic.

Answer 2

Hint: $z$ is algebraic over $K$ means the $K\!$-algebra $K[z]$ is a finite dimensional $K$-vector space. Observe that $K[z+z^2]$ is a subalgebra of $K[z]$.