# Algebraic Field Extension and Degree of Minimal Polynomial

I'm not sure how to solve the following tasks:

Let $$K\subset M$$ be a field extension of a field $$K$$ and $$x,y\in M$$ are algebraic over $$K$$. Let $$G(T_1, T_2)\in K[T_1,T_2]$$ and $$z=G(x,y)\in M$$.

1) Show that $$z$$ is algebraic over $$K$$.

2)Let $$f_x(X), f_y(X), f_z(X) \in K[X]$$ be the minimal polynomials of $$x,y$$ and $$z$$. Show that $$deg(f_z) \le deg(f_x)*deg(f_y)$$.

English isn't my first language and I'm not sure if I got the right translations, so I'll include our definitions:

A field extension of a field $$K$$ is a ring homomorphism $$a:K \to L$$ where $$L$$ is also a field.

Let $$K \subset L$$ be a field extension. An element $$x \in L$$ is algebraic over $$K$$ if there exists a $$f(T) \in K[T]$$, $$f\neq 0$$, with $$f(x)=0$$.

Thanks in advance for any help.

Since $$x,y \in M$$ are algebraic, the extension $$K \subseteq K(x,y)$$ is finite. Hence, $$K(x,y)$$ is an alegbraic extension over $$K$$. Therefore, $$z\in K(x,y)$$ is algebraic over $$K$$. Now, $$K \subseteq K(z) \subseteq K(x,y)$$ are finite extensions. Then, the tower lemma implies $$\deg(f_z)=[K(z): K] \leq [K(x,y):K]\leq \deg(f_x)\deg(f_y).$$
• Hmmm, actually one does not necessarily need the tower law. The only fact we need is that $[K(z):K] \leq [K(x,y):K]$ which I believe you can prove. – user512346 Nov 25 '18 at 2:00