Let $L|K$ be a finite field extension. Let $G=\text{Gal}\ (L|K).$ Let us consider the set $$\text{Fix}_{G}\ (L):= \left \{x \in L\ |\ \sigma(x)=x,\ \text {for all}\ \sigma \in G \right \}.$$ Then $\text{Fix}_{G}\ (L)$ is a subfield of $L$ containing $K.$ If $\text{Fix}_{G}\ (L) = K$ can we say that $L|K$ is a Galois extension?
I have shown the converse of this result which states that "If $L|K$ be a finite Galois extension then $\text{Fix}_{G}\ (L) = K.$"
What I know about finite Galois extension are as follows $:$
$(1)$ Theorem of primitive element for finite Galois extension which states that "Every finite Galois extension $L|K$ is simple i.e. $\exists$ $x \in L$ such that $x$ is a primitive element of $L$ over $K$ or in other words $L=K[x]=K(x).$"
$(2)$ Galois correspondence theorem "if $L|K$ be a finite Galois extension then $L$ is finite Galois extension of any intermediary fields between $K$ and $L.$ Moreover there are inclusions reversing maps between the set of all intermediary fields between $K$ and $L$ and the set of all subgroups of $G=\text{Gal}\ (L|K)$ which are inverses of each other i.e. there exists a one to one correspondence between these two sets."
Now ho do I deduce the required result (if it is at all true) with the help of above results or theorems? What I observed is that if we can prove that $\# \text{Gal}\ (L|K) = [L:K]$ then we are through. How do I prove that? Any help will be highly appreciated.
Thank you very much.