# Composite of two Galois extensions

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!

• Your answer assumes that $L_1$ and $L_2$ are Galois over not just $K$, but also over $L_1\cap L_2$. – user416426 Apr 13 '17 at 6:24
• What if I write $L_{1}=K(\alpha_{1}, \dots , \alpha_{r})$ and $L_{2}=K(\alpha_{r+1}, \dots ,\alpha_{n})$ where $\alpha_{1}, \dots ,\alpha_{r}$ are the roots of a separable polynomial $f_{1}$ over $K$ and $\alpha_{r+1}, \dots \alpha_{n}$ are the roots of another separable polynomial $f_{2}$ over $K$. Then $L_{1}L_{2}=K(\alpha_{1}, \dots , \alpha_{n})$ will be the splitting field of the separable polynomial $\frac{f_{1}f_{2}}{gcd_{K[X]}(f_{1},f_{2})^{2}}$, hence Galois. Would this be a better approach? – Raizen Apr 13 '17 at 8:08
• In the denominator above there should be just the GCD, not GCD squared – Raizen Apr 13 '17 at 8:15
• But, @NoahRiggenbach, isn’t it automatically the case that if $K\subset M\subset L$ and $L$ is finite Galois over $K$, that then $L$ is Galois over $M$? After all, $L$ has been gotten by adjoining all roots of a $K$-polynomial, $f$, and $f$ is all the more so an $M$-polynomial as well. – Lubin Apr 13 '17 at 21:57
• Oh, @NoahRiggenbach, I always mentioned it when I taught Galois Theory. – Lubin Apr 13 '17 at 22:38