Fundamental theorem of Galois theory

Statement
Let $$L$$ be a Galois extension of a field $$K$$ (in other words, $$L$$ is algebraic, normal and separable over $$K$$). Suppose $$G$$ is the Galois group of the field extension $$L/K$$: in other words, $$G$$ is the group of field automorphisms of $$L$$ that fix $$K$$ pointwise. Then, consider the following maps:


 * $$\alpha$$ from subgroups of $$G$$ to subfields of $$L$$, where $$\alpha(H)$$ is the set of elements of $$L$$ that are fixed under every element of the subgroup $$H$$
 * $$\beta$$ from subfields of $$L$$ containing $$K$$, to subgroups of $$G$$. For a subfield $$M$$ of $$L$$, $$\beta(M)$$ is the subgroup of $$G$$ comprising those automorphisms that fix every element of $$M$$

(we can think of $$\alpha$$ and $$\beta$$ as the forward and backward map arising from the following Galois correspondence between $$L$$ and $$G$$: $$g \in G$$ is related to $$a \in L$$ iff $$g$$ fixes $$a$$).

Then, the following are true:


 * 1) $$\alpha \circ \beta$$ is the identity map. In other words, for any subfield of $$L$$ containing $$K$$, it is precisely the set of fixed points of the subgroup fixing it.
 * 2) $$\beta \circ \alpha$$ is the identity map. In other words, for any subgroup of $$G$$, it is precisely the subgroup fixing its fixed field in $$L$$
 * 3) We thus get a bijection (via $$\alpha, \beta$$) between subgroups of $$G$$ and subfields of $$L$$ containing $$K$$. This bijection is inclusion-reversing, with the trivial subgroup corresponding to $$L$$ and the whole group corresponding to $$K$$
 * 4) Given a subgroup $$H \le G$$, the dimension of $$L$$ over $$\alpha(H)$$ equals the cardinality of $$H$$
 * 5) A subgroup $$H \le G$$ is a normal subgroup in $$G$$ iff the corresponding field $$\alpha(H)$$ is a normal field extension of $$K$$