Introduction To p-adic Numbers and p-adic Analysis by A. Baker

By A. Baker

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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 × 25 + · · · ) − (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 + · · · ) . (3 + 2 × 25 + 4 × 125 + · · · ) 56 PROBLEMS Problem Set 4 4-1. ; 22n − 1 2n − 1 1 ; n! for p = 2; pn+1 . pn 4-2. Find the radius of convergence of each of the following power series over Qp : Xn ; n!

If the series has no limit we say that it diverges. 3. Taking αn = npn we have m npn sm = n=1 and sn+1 − sn = (n + 1)pn+1 . 29 30 3. SOME ELEMENTARY p-ADIC ANALYSIS This has norm (n + 1)pn+1 p = |n + 1|p pn+1 p 1 , pn+1 which clearly tends to 0 as n −→ ∞ in the real numbers. 1, (sn ) is a Cauchy sequence and therefore has a limit in Qp . In real analysis, there are series which converge but are not absolutely convergent. For example, the series (−1)n /n converges to − ln 2 but 1/n diverges. Our next result shows that this cannot happen in Qp .

Consider the sequence en where en = 66 0 i n xi . i! Show that en is a Cauchy sequence with respect to | |p if (A) p > 2 and |x|p < 1, or (B) p = 2 and |x|2 < 1/2. In either case, does this sequence have a limit in Q? PROBLEM SET 3 55 Problem Set 3 3-1. Let F be any field and let R = F [X] be the ring of polynomials over F on the variable X. Define an integer valued function ordX f (X) = max{r : f (X) = X r g(X) for some g(X) ∈ F [X]}, and set ordX 0 = ∞. Then define N (f (X)) = e− ordX f (x) .

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