4400/6400 PROBLEM SET 3
A sufficient number of problems: 4 for 4400 students, 6 for 6400 students.
3.1) Let
c
and
N >
1 be integers, and let
c
be the class of
c
modulo
N
.
a) Show that
c
is a unit in
Z
/N
Z
if and only if gcd(
c, N
) = 1.
b) Show that #(
Z
/N
Z
)
×
≤
N

1, with equality holding if and only if
N
is prime.
3.2) Let
m
and
b
be real numbers, and consider the line
:
y
=
mx
+
b.
a) Show that the only possibilities for the number of
Q
rational points (
x, y
) on
are: none, exactly one, infinitely many.
b) Suppose
m
and
b
are both rational. Show
has infinitely many rational points.
c) Suppose
m
is rational and
b
is irrational. Show
has no rational points.
d) Suppose
m
is irrational and
b
is rational. Show
has exactly one rational point.
e) What can be said when
m
and
b
are both irrational?
3.3) (Converse of Wilson’s Theorem) Let
N >
1 be such that
(
N

1)!
≡ 
1
(mod
N
)
.
Show that
N
is prime.
3.4)* Let
D
be a squarefree integer which is not 0 or 1, and put
R
D
:=
Z
[
√
D
] =
{
a
+
b
√
D

a, b
∈
Z
}
∼
=
Z
[
t
]
/
(
t
2

D
)
.
Thus
R
D
is the subring of the complex numbers obtained by adjoining
√
D
to
Z
.
a) Show that
R
D
is an integral domain, with fraction field
Q
[
√
D
] =
{
a
+
b
√
D

a, b
∈
Q
}
.
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 Spring '11
 Staff
 Number Theory, Integers, Integral domain, Ring theory, Commutative ring, Principal ideal domain, weak norm

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