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Problem Set 1
1. Prove case 2 of the Division Algorithm
2. (1.51*)
If
ζ
is a root of unity, prove that there is a positive integer
d
with
ζ
d
= 1 such that if
ζ
k
= 1
then
d

k
3. (1.56*)
Let
a
and
b
be integers and let
sa
+
tb
= 1 for
s,t
∈
Z
. Prove that
a
and
b
are relatively
prime.
4. (1.57*)
If
d
= (
a,b
) prove that
a/d
and
b/d
are relatively prime.
5. (1.60)
If
a
and
b
are relatively prime and if each divides an integer
n
, prove that their product
ab
also divides
n
.
6. (1.73)
Deﬁnition
If
p
is prime, deﬁne the
p
adic norm
of a rational number
a
as follows: if
a
6
= 0,
then
a
=
±
p
e
p
e
1
1
···
p
e
n
n
, where
p
,
p
1
,
...
,
p
n
are distinct primes, and we set

a

p
=
p

e
; if
a
= 0 set

a

p
= 0. Deﬁne the
p
adic metric
by
δ
p
(
a,b
) =

a

b

p
.
(a) For all rationals
a
and
b
, prove that

ab

p
=

a

p

b

p
and

a
+
b

p
≤
max
{
a

p
,

b

p
}
(b) For all rationals
a
and
b
, prove
δ
p
(
a,b
)
≥
0 and
δ
p
(
a,b
) = 0 if and only if
a
=
b
.
(c) For all rationals
a
and
b
prove that
δ
p
(
a,b
) =
δ
p
(
b,a
)
(d) For all rationals
a
,
b
and
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This note was uploaded on 04/19/2011 for the course MATH 21373 taught by Professor Johnson during the Spring '11 term at Carnegie Mellon.
 Spring '11
 Johnson
 Algebra, Division, Integers

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