By Andrew Baker

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Extra resources for An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes]

Sample text

N=0 Of course, this is just the binomial series for (1 − 2)x in Q2 . The exponential and logarithmic series. In real and complex analysis the exponential and logarithmic power series exp(X) = log(X) = ∞ ∑ Xn n! n=0 ∞ ∑ , (−1)n−1 n=1 Xn n are of great importance. We can view each of these as having coeﬃcients in Qp for any prime p. The ﬁrst issue is to determine the p-adic radius of convergence of each of these series. Further details on this material can be found in [5]. 11, the p-adic radii of convergence of the p-adic power series expp (X) = ∞ ∑ 1 n X , n!

1 and x≡0, consider p sn = 1 + x + x2 + · · · + xn−1 ∈ Z. What element of Z/pn does sn represent? (b) Let p be an odd prime. Let n 0, x≡0 and a be an integer such that 2a≡n 1. Show that p rn = 1 + p ∑ 1 k n−1 ( ) 2k (a2 x)k k satisﬁes the equation (rn )2 (1 − x)≡n 1. p For p = 2, show that this equation holds if x≡0. 8 55 Problem Set 2 2-1. Use Hensel’s Lemma to solve each of the following equations: X 2 + 6 ≡ 0; (i) 625 X + X + 8 ≡ 0. B. 2401 = 74 . 2-2. |3 . 2-3. Let p be a prime and n > 0. ) = 1 + p + · · · + pn−1 .

A) For a prime p, n (ii) Z/27, (iii) Z/10. 1 and x≡0, consider p sn = 1 + x + x2 + · · · + xn−1 ∈ Z. What element of Z/pn does sn represent? (b) Let p be an odd prime. Let n 0, x≡0 and a be an integer such that 2a≡n 1. Show that p rn = 1 + p ∑ 1 k n−1 ( ) 2k (a2 x)k k satisﬁes the equation (rn )2 (1 − x)≡n 1. p For p = 2, show that this equation holds if x≡0. 8 55 Problem Set 2 2-1. Use Hensel’s Lemma to solve each of the following equations: X 2 + 6 ≡ 0; (i) 625 X + X + 8 ≡ 0. B. 2401 = 74 . 2-2.