Galois group of the composite of a Galois extension with an arbitrary extension

Let \(F/k\) be a finite Galois extension and \(L/k\) an arbitrary extension, with \(F\) and \(L\) subfields of a common field. Then \(FL/L\) is Galois, and restriction \(\sigma \mapsto \sigma|_F\) is an isomorphism \[ \Gal(FL/L) \cong \Gal(F / F\intersect L) \leq \Gal(F/k). \]