Discrete Valuation

Discrete Valuation is a function ν : A & # x2192; FROM & # x222A; { & # x221E; } {\displaystyle \nu :A\to \mathbb {Z} \cup \{\infty \}}

where A {\displaystyle A} - alternating ring,

meeting the requirements: ν ( x ⋅ Y ) = ν ( x ) + ν ( Y ) {\ displaystyle \ now (x \ cdot y) = \ now (x) + \ now (y)} ν ( x + Y ) & # x2A7E; m i n { ν ( x ) , ν ( Y ) } {\displaystyle \nu (x+y)\geqslant \mathrm {min} {\big \{}\nu (x),\nu (y){\big \}}} ν ( x ) = & # x221E; & # x27FA; x = 0 {\displaystyle \nu (x)=\infty \iff x=0} .

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