I'm reading Lang's Algebraic Number Theory and the Corollary 1 in page 39 says
Let $K$ be a number field and $E$ a finite extension, of degree $n$. Let $v\in M_K$ (the set of all archimedian absolute values and $\mathfrak{p}$-adic valuations) and for each absolute value $w$ on $E$ extending $v$, let $n_w$ be the local degree, $n_w=[E_w:K_v]$. Then $\sum_{w\mid v}n_w=n$.
But I also notice that $E_w=EK_v$, which is independent from the choice of $w$. Thus each $n_w$ are all the same, seems weird. Is this true?
If $E/K$ is Galois then $E_w\cong E_{w'}$ for two places above the same $v$ of $K$. Try with $E=\Bbb{Q}[i],K_v=\Bbb{Q}_5$ and $w,w'$ the places corresponding to the prime ideals $(2+i),(2-i)$.
If $E/K$ is non-Galois then $E K_v$ may depend on your chosen embedding $E\to \overline{K_v}$. For example $x^3-3=(x-1)(x^2+x+1) \bmod 2$ and $$\varprojlim \Bbb{Z}[3^{1/3}]/(2,3^{1/3}-1)^n\cong \Bbb{Z}_2,\qquad \varprojlim \Bbb{Z}[3^{1/3}]/(2,3^{2/3}+3^{1/3}+1)^n\cong \Bbb{Z}_2[\zeta_3]$$ are two different embeddings $\Bbb{Z}[3^{1/3}]\to \overline{\Bbb{Q}_2}$.
Note that due to simple-multiplicites and Hensel lemma the factorization $x^3-3=(x-1)(x^2+x+1) \bmod 2$ lifts to a factorization $x^3-2 = (x-\alpha)(x^2+bx+c)\in \Bbb{Z}_2[x]$ and the two embeddings and places correspond to $$\Bbb{Q}[x]/(x^3-3)\to \Bbb{Q}_2[x]/(x-\alpha),\qquad \Bbb{Q}[x]/(x^3-3)\to \Bbb{Q}_2[x]/(x^2+bx+c)$$