The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
This is an arithmetic question, in that by the normal basis theorem one knows that L is a free K[G]-module of rank 1.
The answer is yes, as one sees by identifying it with Q(ζ) where In fact all the subfields of the cyclotomic fields for p-th roots of unity for p a prime number have normal integral bases (over Z), as can be deduced from the theory of Gaussian periods (the Hilbert–Speiser theorem).
In terms of the discriminant D of L, and taking still K = Q, no prime p must divide D to the power p. Then Noether's theorem states that tame ramification is necessary and sufficient for OL to be a projective module over Z[G].
It leaves the question of the gap between free and projective, for which a large theory has now been built up.
[1] Many objects that arise in number theory are naturally Galois representations.
An important family of examples are the ℓ-adic Tate modules of abelian varieties.
These are the continuous finite-dimensional linear representations of GK on complex vector spaces.
For example, via rK the continuous complex characters of WK are in bijection with those of CK.
If K is a local field of residue characteristic p ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations of WK.
These latter have the nice feature that the continuity of r is only with respect to the discrete topology on V, thus making the situation more algebraic in flavor.