Valuation over the algebraically closed field of rational number

Question

How do we define the valuation over the algebraically closed field of rational numbers say $\bar{\mathbb Q}$ as an extension of the valuation of $\mathbb Q$ ?

Answer 1

For any finite Galois extension $K/\mathbb{Q}_p$, there is a unique extension of the norm that respects the $p$-adic norm on $\mathbb{Q}_p$, and this is Galois-invariant. Therefore, it must be given by $|x|_K = |Norm(x)|_p^{1/[K:\mathbb{Q}_p]}$. By uniqueness, if we have a tower of field extensions $L/K/\mathbb{Q}_p$, then restricting the norm on $|\cdot |_L$ to $K$ gives $|\cdot |_K$. Since any element of $\overline{Q}$ lives in a finite Galois extension of $\mathbb{Q}_p$, this gives a way to extend the norm to all of $\overline{Q}$.