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n
1-4. Find a generator for the cyclic group of units (Z/n)× in each of the following rings: (i)
Z/23, (ii) Z/27, (iii) Z/10.
1-5. a) For a prime p, n 1 and xa"0, consider
p
sn = 1 + x + x2 + · · · + xn-1 " Z.
What element of Z/pn does sn represent?
n
b) Let p be an odd prime. Let n 0, xa"0 and a be an integer such that 2aa"1. Show that
p p
2k
rn = 1 + (a2x)k
k
1 k n-1
satisfies the equation
n
(rn)2(1 - x)a"1.
p
For p = 2, show that this equation holds if xa"0.
8
53
54 PROBLEMS
Problem Set 2
2-1. Use Hensel s Lemma to solve each of the following equations:
(i) X2 + 6 a" 0;
625
(ii) X2 + X + 8 a" 0.
2401
N.B. 2401 = 74.
2-2. Determine each of the following numbers:
ord3 54, ord5(-0.0625), ord7(-700/197), | - 128/7|2, | - 13.23|3, |9!|3.
2-3. For a " Q what condition on |a|p must be satisfied to ensure that the equation 5x2 = a to
have a solution (i) in Z, (ii) in Q?
2-4. Let p be a prime and n > 0.
a) Show that ordp(pn!) = 1 + p + · · · + pn-1.
b) When 0 a p - 1, show that
ordp(apn!) = a(1 + p + · · · + pn-1).
c) Let r = r0 + r1p + · · · + rdpd, where 0 rk p - 1 for each k, and set ±p(r) = ri. Show
0 i d
that
r - ±p(r)
ordp(r!) = .
p - 1
Use this to determine |r!|p.
2-5. a) Show that
1
|x|p = ,
|x|
p
where the product is taken over all primes p = 2, 3, 5, . . . and x " Q.
b)If x " Q and |x|p 1 for every prime p, show that x " Z.
2-6. Let p be a prime and x " Q. Consider the sequence en where
xi
en = .
i!
0 i n
Show that en is a Cauchy sequence with respect to | |p if (A) p > 2 and |x|p
and |x|2
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) = Xrg(X) for some g(X) " F [X]},
and set ordX 0 = ". Then define
N(f(X)) = e- ordX f(x).
Prove that ordX satisfies the conditions of Proposition 2.4 with ordX in place of ordp. Hence
deduce that N satisfies the conditions required to be a non-Archimedean norm on R.
3-2. Which of the following are Cauchy sequences with respect to the p-adic norm | |p where p
is a given prime?
n
(a) n!, (b) 1/n!, (c) xn (this depends on x), (d) ap (this depends on a), (e) ns for s " Z (this
depends on s).
In each case which is a Cauchy sequence find the limit if it is a rational number.
3-3. Let f(X) " Z[X] and let p be a prime. Suppose that a0 " Z is a root of f(X) modulo p
(i.e., f(a0)a"0). Suppose also that f (a0) is not congruent to 0 mod p. Show that the sequence
p
(an) defined by
an+1 = an - uf(a0),
where u " Z satisfies uf (a0)a"1, is a Cauchy sequence with respect to | |p converging to root of
p
f in Qp.
3-4. Let p be a prime with pa"1.
4
a) Let c " Z be a primitive (p - 1)-st root of 1 modulo p. By considering powers of c, show that
there is a root of X2 + 1 modulo p.
b) Use Question 3-2 to construct a Cauchy sequence (an) in Q with respect to | |p such that
1
a2 + 1 p
n
pn
c) Deduce that there is a square root ± of -1 in Q5.
d) For p = 5 find ±1 " Q so that
1
2
|±1 + 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 an 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:
± = (3/5 + 2 + 4 × 5 + 0 × 25 + 2 × 25 + · · · ) - (4/5 + 3 × 25 + 3 × 125 + · · · ),
² = (1/25 + 2/5 + 3 + 4 × 5 + 2 × 25 + 2 × 125 + · · · ) × (3 + 2 × 5 + 3 × 125 · · · ),
(5 + 2 × 25 + 125 + · · · )
³ = .
(3 + 2 × 25 + 4 × 125 + · · · )
56 PROBLEMS
Problem Set 4
4-1. Discuss the convergence of the following series in Qp:
1 22n - 1 pn+1
n!; ; for p = 2; .
n! 2n - 1 pn
4-2. Find the radius of convergence of each of the following power series over Qp:
n
Xn Xp Xn
; pnXn; ; nkXn with 0 k " Z fixed; n!Xn; .
n! pn n
4-3. Prove that in Q3,
"
32n " 32n
= 2 .
42nn 4nn
n=1 n=1
4-4. For n 1, let
X(X - 1) · · · (X - n + 1)
Cn(X) =
n!
and C0(X) = 1; in particular, for a natural number x,
x
Cn(x) = .
n
a) Show that if x " Z then Cn(x) " Z.
c) Show that if x " Zp then Cn(x) " Zp.
c) If x " Zp and ±n " Qp, show that the series
"
±nCn(x),
n=0
converges if and only if
lim ±n = 0.
n’! "
"
d) For x " Z, determine Cn(x)pn.
n=0
PROBLEM SET 5 57
Problem Set 5
5-1. a) Let ±n be a series in Qp. Prove that the p-adic Ratio Test is valid, i.e., ±n
converges if
±n+1
» = lim
n-’! " ±n p
exists and »
b) If ³nXn is a power series in Qp, deduce that the p-adic Ratio Test for Power Series is
valid, i.e., if
³n
» = lim
n-’! "
³n+1 p
exists then ³nXn converges if |x|p ».
Use these tests to determine the radii of convergence of the following series.
pn+1 n Xn pn
Xn; n!Xn; pnXn; ; pnXn; Xn.
p pn n
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