UMMARY OF
11/12 L
ECTURE
We began class with a quick refresher on the geometry of the complex plane,
C
. Most im
portantly, we discussed the magnitude of a complex number: it is the physical distance from
the point representing that number, to the origin. Thus,

i

= 1
, despite the temptation to
say that the ‘distance’ from
i
to 0 is
i
. It is easy to prove (by Pythagorean theorem!) that

a
+
bi

=
√
a
2
+
b
2
.
We then returned to our theorem that
Z
×
p
is cyclic. Recall that last time we showed that any
ﬁnite nice group
G
must be cyclic – here ‘nice’ means that for every
n
≥
1
,
x
n
= 1
has at most
n
distinct solutions in
G
. Thus, to prove our theorem it remains only to show that
Z
×
p
is nice. We will prove rather more:
Theorem 1.
Suppose
f
(
x
)
is a monic polynomial with integer coefﬁcients, and that
deg
f
≥
1
.
Then the number of distinct elements
x
∈
Z
p
satisfying
f
(
x
)
≡
0 (
mod
p
)
is at most the degree of
f
.
[Recall that
f
has degree
n
(written
deg
f
=
n
) if the highest power of
x
appearing in
f
(
x
)
is
x
n
.
f
is
monic
if the coefﬁcient of the highestdegree term is 1. Thus a monic polynomial of
degree
n
looks like
f
(
x
) =
x
n
+
c
1
x
n

1
+
c
2
x
n

2
+
···
+
c
n

1
x
+
c
n
.]
It will be convenient to deﬁne
Z
[
x
]
to be the collection of
all
polynomials with integer coefﬁ
cients.
Proof.
Let
Z
p
(
f
) =
{
x
∈
Z
p
:
f
(
x
)
≡
0 (
mod
p
)
}
. We need to show that

Z
p
(
f
)
 ≤
deg
f
for every monic polynomial
f
∈
Z
[
x
]
with
deg
f
≥
1
. We do this by induction:
deg
f
= 1
:
In this case, we must have
f
(
x
) =
x

a
for some integer
a
. It is a good exercise to show that
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 Fall '10
 GideonMaschler
 Geometry, Complex number, deg, zp, good exercise

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