Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value. == Theorem statement == An absolute value on the rational numbers is a function | ⋅ | ∗ : Q → R {\displaystyle |\cdot |_{*}:\mathbb {Q} \to \mathbb {R} } satisfying for all x , y ∈ Q {\displaystyle x,y\in \mathbb {Q} } | x | ∗ ≥ 0 {\displaystyle |x|_{*}\geq 0} , with equality if and only if x = 0 {\displaystyle x=0} | x y | ∗ = | x | ∗ | y | ∗ {\displaystyle |xy|_{*}=|x|_{*}|y|_{*}} | x + y | ∗ ≤ | x | ∗ + | y | ∗ {\displaystyle |x+y|_{*}\leq |x|_{*}+|y|_{*}} Two absolute values | ⋅ | {\displaystyle |\cdot |} and | ⋅ | ∗ {\displaystyle |\cdot |_{*}} on the rationals are defined to be equivalent if they induce the same topology; this can be shown to be equivalent to the existence of a positive real number λ ∈ ( 0 , ∞ ) {\displaystyle \lambda \in (0,\infty )} such that | x | ∗ = | x | λ {\displaystyle |x|_{*}=|x|^{\lambda }} for all rational x {\displaystyle x} .