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I guess everybody knows this, I am just mentioning it: in Atiyah-MacDonald's Commutative Algebra, the proof for the extension of valuation is done by using maps $f$ from the original ring A (in our case $\mathbb{Z}_p$, corrected to(p)) to an algebraic closure (say $\mathbb{C}$ or as Laurent Berger mentioned $\overline{\mathbb{Q}_p}$), \overline{\mathbb{Q}_p}$, or$\overline{\mathbb{F}_p}$in our case), and then for any$a\not\in A$, there is a way to decide whether we can extend the valution to$A[a]$or$A[a^{-1}]$, using the maps previously defined. To me, this method seems a little bit more explicit, which is actually not. 2 added 40 characters in body I guess everybody knows this, I am just mentioning it: in Atiyah-MacDonald's Commutative Algebra, the proof for the extension of valuation is done by using maps$f$from the original ring A (in our case$\mathbb{Z}_p$) \mathbb{Z}_p$, corrected to (p)) to an algebraic closure (say $\mathbb{C}$ or as Laurent Berger mentioned $\overline{\mathbb{Q}_p}$), and then for any $a\not\in A$, there is a way to decide whether we can extend the valution to $A[a]$ or $A[a^{-1}]$, using the maps previously defined. To me, this method seems a little bit more explicit, which is actually not.
I guess everybody knows this, I am just mentioning it: in Atiyah-MacDonald's Commutative Algebra, the proof for the extension of valuation is done by using maps $f$ from the original ring A (in our case $\mathbb{Z}_p$) to an algebraic closure (say $\mathbb{C}$ or as Laurent Berger mentioned $\overline{\mathbb{Q}_p}$), and then for any $a\not\in A$, there is a way to decide whether we can extend the valution to $A[a]$ or $A[a^{-1}]$, using the maps previously defined. To me, this method seems a little bit more explicit, which is actually not.