Let $ K =\mathbb{F}_{2}(X,Y) $, $ L=K[Z]/(Z^{2}+X) $ and $ M=L[T]/(T^{2}+Y) $. Show that for every $ a \in M \setminus K$, we have that the degree of $ a $ over $ K $ i.e. the degree of its minimal polynomial over $ K $ is $ 2 $ and that the field extension $ K \subseteq M $ is not simple.
I showed that $ M/K $ has degree $ 4 $, a basis for it being $ \{1,z,t,zt\}$, where $ z $ and $ t $ are the images of $ Z $ in $ L $ and of $ T $ in $ M $ respectively, hence if we show that every element in $ M \setminus K $ has degree $ 2 $, the fact that the extension $ M/K $ is not simple follows easily, but I can't think of a way of proving the first statement, except by brute force and maybe finding for every such $ a $ an irreducible polynomial over $ K $ of degree $ 2 $ that has $ a $ as a root, though I would like a clever way of showing this.
I would appreciate any help. Thank you!