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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.

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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 $$ \deg(f_z)=[K(z): K] \leq [K(x,y):K]\leq \deg(f_x)\deg(f_y). $$

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  • $\begingroup$ Thank you, I couldn't find the tower lemma in my lecture notes but maybe we'll get there in the next lecture on Monday. $\endgroup$ – Anzu Nov 25 '18 at 1:57
  • $\begingroup$ 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. $\endgroup$ – user512346 Nov 25 '18 at 2:00

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