Class group embedding in coprime extension Let $L/K$ be an extension of number fields of degree $n$. Assume that the class group of $K$ has order $h$. Prove that if $(h,n)=1$ the map $Cl(K)\rightarrow Cl(L)$, given by $I\rightarrow I\mathcal O_{L}$ is then an injection. I tried to use the ram-rel formulas, but still I can't get to make use of the coprimality assumption, because we don't even know that $L/K$ is Galois.

Let I in $O_K$, by the norme in L / K we obtained $N(IO_L)=I^n$

so if class of I is trivial in cl(L), then by nome its nth power is also trivial, but n is coprime to the order of class I, so itself is trivial.