Let $K=\Bbb Z_p(U,V)$, the field of rational functions in two variables $U$ and $V$ over $\Bbb Z_p,$ be taken as the base field. Now consider the polynomial $f(X)=(X^p-U)(X^p-V) \in K[X].$ Then by Kronecker's theorem $\exists$ a field $L'$ such that $f(X)$ splits completely into linear factors in $L'[X].$ But that will imply $X^p-U$ and $X^p-V$ split completely into linear factors in $L'[X].$ Cosequently both these polynomials have a root in $L'.$ Let $u$ and $v$ be some zeros of $X^p-U$ and $X^p-V$ respectively. Then $u^p=U$ and $v^p=V.$ Now using the fact that $\text{Char}\ (L')=p$ we find that $$X^p-U=X^p-u^p=(X-u)^p.$$ So $u$ is the only zero of $X^p-U$ with multiplicity $p.$ Similarly we can show that $v$ is the only zero $X^p-V.$ Let $V_{L'} (f(X))$ the set of zeros of $f(X)$ in $L'.$ Then we have $$V_{L'} (f(X)) = \{u,v \}.$$
Let $L=K(u,v)$ be the smallest subfield of $L'$ containing $K,u$ and $v.$ Then we have $$V_L(f(X))=\{u,v \}.$$ Since both $U$ and $V$ are primes in $\Bbb Z_p(U,V)$ so both the polynomials $X^p-U$ and $X^p-V$ are irreducible in $K[X].$ So they cannot have any root inside $K.$ Therefore $u,v \notin K.$ Therefore the degree of extension $[L:K] > 1.$ Now let $G= \text {Gal}\ (L|K).$ Let $\sigma \in G.$ Since $X^p-U$ and $X^p-V$ are respectively the minimal polynomials of $u$ and $v$ over $K$ it follows that $\sigma (u) \in V_L(X^p-U) = \{u \}$ and $\sigma (v) \in V_L(X^p-V) = \{v \}.$ But this implies that $\sigma(u)=u$ and $\sigma(v)=v.$ Since $\sigma$ is completely determined by the pair $(\sigma(u),\sigma(v))=(u,v).$ It follows that $\sigma = \text{id}_L.$ This shows that $G$ is a trivial group i.e. $\#\ G=1 < [L:K].$ Hence $L$ is not a Galois extension of $K.$
Now my question is can we not show the non Galois extension just by taking $K=\Bbb Z_p(U), f(X)= X^p-U$ and $L=K(u)$? Then also I believe $L$ is not a Galois extension of $K.$ Right? But then what is necessity of adjoing two variables to $\Bbb Z_p$ to demonstrate an example of a non Galois extension as given in the lecture notes provided by our instructor?
Any help in this regard will be highly appreciated. Thank you very much.