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For p a given prime number, define the p-adic norm | * |p as follows on...

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Advance Math

For p a given prime number, define the p-adic norm | * |p asfollows on Q: Given q in Q, we can write it as a product q =(p^m)(a/b) with a,b integers which are not divisible by p, and m aninteger which is uniquely determined by q (check that m is indeeduniquely determined by q). Then define |q|p = p^(-m).

Check that Q with distance dp(q1,q2) = |q1 - q2|p is a metricspace (here q1-q2 just means the usual operation of subtraction inQ). Show moreover
that |q1 + q2|p <= max(|q1|p, |q2|p).

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