Sorry if this question is too basic. It is from Fröhlich and Taylor's "Algebraic Number Theory".
Let $E/F$ be a finite Galois extension of fields, with $G=Gal(E/F)$, and let $K$ and $L$ be two subfields of $E$, containing $F$, such that $K/F$ and $L/F$ are both Galois. Let $M=Gal(E/K)$ and $N=Gal(E/L)$ be normal subgroups of $G$. Suppose ${\gamma_1,\ldots,\gamma_n}$ is a transversal for $MN$ in $G$, with $n=[G:MN]$. If $C$ is the compositum $KL$ in $E$, how can I show the map $$ k\otimes l\mapsto (k^{\gamma_1}l,\ldots,k^{\gamma_n}l) $$
induces an isomorphism between $K\otimes_F L$ and $\prod_{i=1}^n C$?
It is clear to me that this map is an $F$-algebra homomorphism, and that they both have the same dimension over $F$. Thus surjectivity, or injectivity, would be enough. I have not been able to figure out what the idempotents of $\prod_{i=1}^n C$ should look like in $K\otimes_F L$, so I have not been able to show surjectivity. Meanwhile, I think injectivity should be easier to show, because if we have $$ k_1\otimes l_1 + \cdots + k_m\otimes l_m\mapsto 0,$$ then we get a system of equations $$ k_1l_1 + \cdots + k_ml_m=0$$ $$ \cdots$$ $$ k_1^{\gamma_n}l_1 + \cdots k_m^{\gamma_n}l_m=0.$$ Summing up the columns, I get $$ \sum_{i=1}^m(\sum_{j=1}^n k_i^{\gamma_j})l_i=0 $$ and all this is happening in $L$. But I can't seem to finish this argument. Any help would be greatly appreciated.