# abstract algebra – The norm of the field commutes with the embeddings

Let $$L / K$$ to be a finite extension of number fields. Let $$sigma$$ to be an integration of $$K$$ in $$mathbb {C}$$. Choose an integration $$sigma & # 39;$$ of $$L$$ in $$mathbb {C}$$ extension $$sigma$$. then $$sigma & # 39; (L)$$ is a finite extension of $$sigma (K)$$ as subfields of $$mathbb {C}$$. I must show that the following diagram commutes.
$$require {AMScd}$$
$$begin {CD} L @> sigma & # 39; >> sigma & # 39; (L) \ @ VVN_ {L / K} V @VVN _ { sigma (L) / sigma (K)} V \ K @> sigma >> sigma (K) end {CD}$$

My attempt: As any incorporation of fields is a monomorphism, $$K cong sigma (K)$$ and $$L cong sigma (L)$$. Also, since $$sigma: L to mathbb {C}$$ s & # 39; extends $$sigma: K to mathbb {C}$$, we have $$sigma & # 39; _ {K} = sigma$$. So, $$(L: K) = ( Sigma (L): Sigma (K)) = n$$ and $$sigma (K) = sigma & # 39; (K)$$.

Let $$mu_1, ldots, mu_n$$ Be the $$(L: K) = n$$ boarding $$L$$ in $$mathbb {C}$$ what fix $$K$$ on point. Then for $$alpha in L$$, we have
begin {align *} N_ {L / K} ( alpha) & = mu_1 ( alpha) cdots mu_n ( alpha) end {align *}

Let $$lambda_1, ldots, lambda_n$$ Be the $$( sigma (L): sigma (K)) = n$$ boarding $$sigma & # 39; (L)$$ in $$mathbb {C}$$ what fix $$sigma (K)$$ on point. Then for $$beta in sigma & # 39; (L)$$, we have
begin {align *} N _ { sigma # (L) / sigma (K)} ( beta) & = lambda_1 ( beta) cdots lambda_n ( beta) end {align *}

How to proceed further? Indices are welcome.