OSTROWSKI’S THEOREM FOR Q(i)
نویسنده
چکیده
We will extend Ostrowki’s theorem from Q to the quadratic field Q(i). On Q, every nonarchimedean absolute value is equivalent to the p-adic absolute value for a unique prime number p, and the archimedean absolute values are all equivalent to the usual absolute value on Q. We will see a similar thing happens in Q(i): any non-archimedean absolute value is associated to a prime in Z[i] (unique up to unit multiple) and any archimedean absolute value is equivalent to the complex absolute value |a+ bi|∞ = √ a2 + b2. First we need some background about Z[i]. For α = a+bi in Z[i] (a, b ∈ Z), set the norm of α to be N(α) = a + b,
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