MATH 223A — HOMEWORK 1

Problem 1 (The p-adic exponential and logarithm). Let (K, | · |) be a complete non-Archimedeanvalued field.

(1) For a0, a1, a2, . . . ,∈ K× define r = (lim sup |an|1/n)−1 ∈ [0,∞]. Prove that∑

anxn convergesfor x ∈ K if |x| < r and does not converge if |x| > r. (Recall that an infinite sum in anon-Archimedean field converges if and only if the summands tend to zero.) Prove that if thesum converges for one x0 with |x0| = r then it converges for all x with |x| = r. Also provethat if r > 0 then

∑anxn = 0 for all x near zero if and only if an = 0 for all n.

(2) Assume from now on that K has characteristic zero and that its residue field k has charac-teristic p. Replace | · | with a suitable power to assume that |p| = p−1. Prove that the powerseries

logp(1 + x) =∞∑

n=1

(−1)n+1 xn

n

converges if and only if |x| < 1, and that logp((1+x)(1+y)) = logp(1+x+y +xy) is equal tologp(1+x)+logp(1+y) whenever |x|, |y| < 1. Prove that | logp(1+x)| = |x| if |x| < p−1/(p−1).

(3) For x ∈ K define ordp(x) ∈ R by |x| = p− ordp(x). Still assuming that char(K) = 0, prove thatordp(n!) = (n − Sn)/(p − 1) for a positive integer n, where Sn is the sum of the digits in thebase-p decimal expansion of n. Conclude that the formal power series

expp(x) =∞∑

n=0

xn

n

converges if and only if |x| < p−1/(p−1), and that | expp(x) − 1| = |x| for such x. Prove thatexpp(x + y) = expp(x) expp(y) whenever |x|, |y| < p−1/(p−1).

(4) For r < 1 prove that the open disc {|x − 1| < r} is a multiplicative subgroup of K×. Showthat x 7→ logp(x) and x 7→ expp(x) define inverse isomorphisms between the multiplicativesubgroup {|x − 1| < p−1/(p−1)} ⊂ K× and the additive subgroup {|x| < p−1/(p−1)} ⊂ K.Show that both maps are continuous with respect to the topologies induced by K.

(5) Taking K = Qp, prove that logp maps the multiplicative group 1 + pZp homeomorphicallyonto additive group pZp if p > 2 and that it maps 1 + 4Z2 homeomorphically onto 4Z2 ifp = 2. Conclude that all elements of 1 + 8Z2 have square roots in Q2.

(6) For any prime p, describe all quadratic extensions of Qp.

Problem 2 (nth powers are open).

(1) Let K be a non-Archimedean local field with residue characteristic p. Use Hensel’s lemmato prove that if n is a nonzero integer then (K×)n is open with finite index in K× whenp - n. When char(K) = 0 use the p-adic logarithm to prove that the same result is true for anynonzero integer n. Deduce that any subgroup of finite index in K× is open when char(K) = 0.

(2) Let K be a local field of positive characteristic p. Using an isomorphism K ∼= k((t)), show that(K×)p is closed but not open in K×. Also prove that there is a finite-index subgroup of K×

that is not open. [Hint: prove that K×/(K×)p is an Fp-vector space of infinite dimension.]

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