Let $ L/K $ be a finite extension of fields and $ L_{1},L_{2} $ two intermediate fields that are Galois over $ K $. Is the composite field $ L_{1}L_{2} $ (i.e. the smallest subfield of $ L $ that contains both $ L_{1} $ and $ L_{2}$) Galois over $ K $?
My thought would be that this indeed is true since as $ L_{1} $ is Galois over $ K $, it is the splitting field of a family of separable polynomials $ \{f_{i} \} _{i \in I} $ over $ K $ and therefore $ L_{1}L_{2} $ is the splitting field of the same family of polynomials over $ L_{2} $. On the other hand, as $ L_{2} $ is Galois over $ K $, $ L_{2} $ is also the splitting field of a family $ \{g_{j}\}_{j \in J} $ of separable polynomials over $ K $ so $ L_{1}L_{2} $ is the splitting field of $ \{f_{i} \} _{i \in I} \cup \{g_{j}\}_{j \in J} $ over $ K $ thus the composite is Galois over $ K $.
Is there anything wrong with my answer? Thank you in advance for any help!