Let $V,V'$ be two discrete valuation rings of a field $K$. Then $A:=V\cap V'$ is a normal domain. Can we say anything more about $A$? For example, can we say that $A$ is Noetherian/one-dimensional? Is $K$ the field of fractions of $A$?
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Yes $K$ is the fraction field. This is because $O_v^\times$ determinates $v$ so given $w\ne v$ we can pick $a\in K,v(a)=0,w(a)>0$ which gives that for any $b\in O_v$ there is $n$ such that $a^n b\in O_v\cap O_w$. – reuns Apr 13 '22 at 18:51
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Thanks. OK, the argument works for rank one valuations. But what happens if $V,V'$ are valuation rings of rank $>1$? Also, any idea about the other two questions? – sagnik chakraborty Apr 14 '22 at 06:02