Math 514b Algebraic Number Theory Spring 1996

Homework assignment 1:

1. Let c > 1 be a real number and F a field. Prove that the function

|f/g|_{∞} = c^{− deg g+deg f} f, g ∈ F[t]

defines a non-archimedean valuation on F(t). What happens if we change the signs (or equivalently, if c < 1)?

2. Prove that if | · | is a non-trivial valuation on a field K, then the field operations are continuous with respect to the valuation topology.

3. Suppose K is complete with respect to a non-archimedean valuation | · |. Prove that the series ∑ a_{n} converges if and only if lim_{n→∞} |a_{n}| = 0.

4. Let | · |_{p} be the p-adic valuation on Q.

a) Compute |n!|_{p} for all positive integers n. (Hint: First consider n = p^{m} and n = ap^{m} for a = 0, …, p − 1.)

b) Determine the radii of convergence of the power series

exp z = ∑_{n=0}^{∞} \frac{z^{n}}{n!}

and

log(1 − z) = − ∑_{n=1}^{∞} \frac{z^{n}}{n}

where z ∈ Q_{p} is a p-adic variable.

5. Define

\binom{1/2}{0} = 1 and \binom{1/2}{n} = \frac{(1/2)(1/2 − 1) ⋯ (1/2 − n + 1)}{n!}

for integers n ≥ 1. Calculus students sometimes learn that if x is a real (or complex) number with |x| < 1, then the series

∑_{n=0}^{∞} \binom{1/2}{n}x^{n}  ∗

converges to a square root of 1 + x.

a) Show that for all odd primes p and all n, |\binom{1/2}{n}|_{p} ≤ 1.

b) Conclude that for odd p and x ∈ Q_{p} with |x|_{p} < 1, the series ∗ converges to a square root of 1 + x.

c) Give an example where the p-adic and real series both converge to a rational number, but the two

limits are different. (!!)

6. In this exercise we consider p-adic expansions of numbers in Q_{p} computed with the uniformiser p and the naive digits: a = ∑_{i=N}^{∞} a_{i}p^{i} with a_{i} ∈ {0, …, p − 1}.

a) Show that the expansion of a ends has only finitely many non-zero terms if and only if a > 0, a ∈ Q, and p^{n}a ∈ Z for large n.

b) Show that the expansion of a is eventually periodic (i.e., a_{i} = a_{i+k} for some k and all sufficiently

large i) if and only if a ∈ Q.

7. Analyze the group Q_{p}^{×}/Q_{p}^{×2} and find a set of coset representatives. What happens when 2 is replaced by a prime ℓ?