EDB — 0XH

Exercises

1. [0XH] Prove these fundamental relation.

   1. \(|1|_ p=1\) and more generally \(|n|_ p≤ 1\) for every nonnull integer \(n\), with equality if \(n\) is not divisible by \(p\).

   2. Given \(n\) nonnull integer, we have that \(|n|_ p=p^{-𝜑_ p(n)}\).

   3. Given \(n,m\) integers, we have that \(𝜑_ p(n+m)≥ \min \{ 𝜑_ p(n),𝜑_ p(m)\} \) with equality if \(𝜑_ p(n)≠ 𝜑_ p(m)\).

   4. Given \(n,m\) nonzero integers, we have that \(𝜑_ p(nm)=𝜑_ p(n)+𝜑_ p(m)\) and therefore \(|nm|_ p=|n|_ p |m|_ p\).

   5. Given \(x=a/b\) with \(a,b\) nonnull integers we have that \(|x|_ p=p^{-𝜑_ p(a)+𝜑_ p(b)}\). Note that if \(a,b\) are coprime, then one of the two terms \(𝜑_ p(a),𝜑_ p(b)\) is zero.

   6. Prove that \(|x y|_ p = |x|_ p |y|_ p\) for \(x,y∈ℚ\).

   7. Prove that \(|x/y|_ p = |x|_ p / |y|_ p\) for \(x,y∈ℚ\) nonzero.

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