While I was reading J.-P. Serre'book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to identify the quotient $\mathfrak{p}_L^{n}/\mathfrak{p}_L^{n+1}$ with the tensor product $\overline{L}\otimes_K\mathfrak{p}_K^n/\mathfrak{p}_K^{n+1}$. Clearly the setting is the tipycal one, K Is a field complete wrt a discrete valuation, $\mathfrak{p}_K$ is its unique prime and $L$ is a finite unramified extension of K. $\overline{L}$ is its residue field. Actually, I cannot see how the tensor product is defined as there are no maps from K to $\overline{L}$ at least when the characteristic of K is different from the one of $\overline{L}$.

While I was reading J.-P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to identify the quotient $\mathfrak{p}_L^{n}/\mathfrak{p}_L^{n+1}$ with the tensor product $\overline{L}\otimes_K\mathfrak{p}_K^n/\mathfrak{p}_K^{n+1}$. Clearly the setting is the typical one, K is a field complete wrt a discrete valuation, $\mathfrak{p}_K$ is its unique prime and $L$ is a finite unramified extension of K. $\overline{L}$ is its residue field. Actually, I cannot see how the tensor product is defined as there are no maps from K to $\overline{L}$ at least when the characteristic of K is different from the one of $\overline{L}$.

While I was reading Serre'book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to identify the quotient $\mathfrak{p}_L^{n}/\mathfrak{p}_L^{n+1}$ with the tensor product $\overline{L}\otimes_K\mathfrak{p}_K^n/\mathfrak{p}_K^{n+1}$. Clearly the setting is the tipycal one, K Is a field complete wrt a discrete valuation, $\mathfrak{p}_K$ is its unique prime and $L$ is a finite unramified extension of K. $\overline{L}$ is its residue field. Actually, I cannot see how the tensor product is defined as there are no maps from K to $\overline{L}$ at least when the characteristic of K is different from the one of $\overline{L}$.

While I was reading J.-P. Serre'book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to identify the quotient $\mathfrak{p}_L^{n}/\mathfrak{p}_L^{n+1}$ with the tensor product $\overline{L}\otimes_K\mathfrak{p}_K^n/\mathfrak{p}_K^{n+1}$. Clearly the setting is the tipycal one, K Is a field complete wrt a discrete valuation, $\mathfrak{p}_K$ is its unique prime and $L$ is a finite unramified extension of K. $\overline{L}$ is its residue field. Actually, I cannot see how the tensor product is defined as there are no maps from K to $\overline{L}$ at least when the characteristic of K is different from the one of $\overline{L}$.