Ostrowski’s theorem

Example 1.3. Let p be a prime number. For any 0 6= a ∈ Q, we can write a = p b c where m, b, c ∈ Z, (bc, p) = 1. Define |a|p = 1 pm and |0|p = 0, then |·|p is a nonarchimedean valuation on Q. Note that for different primes p and q, |·|p and |·|q are not equivalent. For z ∈ C, define |z|∞ = |z| (the usual absolute value). Then |·|∞ is an archimedean valuation on C(thus is not equivalent to |·|p for any p).