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.