By Andrew Baker

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D) For p = 5 ﬁnd α1 ∈ Q so that 1 |α12 + 1|5 < . 3125 3-5. Let R be a ring equipped with a non-Archimedean norm N . Show that a sequence (an ) is Cauchy with respect to N if and only if (an+1 − an ) is a null sequence. Show that this is false if N is Archimedean. 3-6. Determine each of the following 5-adic numbers to within an error of norm at most 1/625: α = (3/5 + 2 + 4 × 5 + 0 × 25 + 2 × 125 + · · · ) − (4/5 + 3 × 25 + 3 × 125 + · · · ), β = (1/25 + 2/5 + 3 + 4 × 5 + 2 × 25 + 2 × 125 + · · · ) × (3 + 2 × 5 + 3 × 125 · · · ), γ= (5 + 2 × 25 + 125 + · · · ) .

8. For the series ∑ nxn , we have |x|p n |nxn |p = |n|p |xn |p which tends to 0 in R if |x|p < 1. So this series certainly converges for every such x. Just as in real analysis, we can deﬁne a notion of radius of convergence for a power series in Qp . For technical reasons, we will have to proceed with care to obtain a suitable deﬁnition. We ﬁrst need to recall from real analysis the idea of the limit superior (lim sup) of a sequence of real numbers. 9. A real number ℓ is the limit superior of the sequence of real numbers (an ) if the following conditions are satisﬁed: (LS1) For real number ε1 > 0, ∃M1 ∈ N such that n > M1 =⇒ ℓ + ε1 > an .

3-10. For n 1, let X(X − 1) · · · (X − n + 1) n! and C0 (X) = 1; in particular, for a natural number x, ( ) x Cn (x) = . n Cn (X) = (a) Show that if x ∈ Z then Cn (x) ∈ Z. (b) Show that if x ∈ Zp then Cn (x) ∈ Zp . (c) If αn ∈ Qp , show that the series ∞ ∑ αn Cn (x), n=0 converges for all x ∈ Zp if and only if lim αn = 0. n→∞ ∑ n (d) For x ∈ Z, determine ∞ n=0 Cn (x)p . ∑ ∑ 3-11. (a) Let αn be a series in Qp . , αn converges if αn+1 λ = lim n−→∞ αn p exists and λ < 1. , if γn λ = lim n→∞ γn+1 p ∑ exists then γn X n converges if |x|p < λ and diverges if |x|p > λ.

### An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes] by Andrew Baker

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