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108
If
x
,
y
∈
R
and
x
±=
0, there are
q
,
r
∈
R
with
y
=
qx
+
r
,
where either
r
=
0o
r
∂(
r
)<∂
(
x
)
. In the second case, it follows that
∂
0
(
r
)<∂
0
(
x
)
,for
m
≥
1
>
0. Hence,
∂
0
is also a degree function on
R
.
3.69
Let
R
be a Euclidean ring with degree function
∂
.
(i)
Prove that
∂(
1
)
≤
∂(
a
)
for all nonzero
a
∈
R
.
Solution.
By the
f
rst axiom in the de
f
nition of degree,
∂(
1
)
≤
∂(
1
·
r
)
=
∂(
r
).
(ii)
Prove that a nonzero
u
∈
R
is a unit if and only if
∂(
u
)
=
∂(
1
)
.
Solution.
If
u
is a unit, then there is
v
∈
R
with
u
v
=
1; therefore,
∂(
u
)
≤
∂(
u
v)
=
∂(
1
)
. By part (i),
∂(
u
)
=
∂(
1
)
.
Conversely, assume that
∂(
u
)
=
∂(
1
)
. By the
“
division algorithm
”
in
R
, there are
q
,
r
∈
R
with 1
=
qu
+
r
, where either
r
=
0or
∂(
r
)<∂
(
u
)
.B
u
t
∂(
u
)
=
∂(
1
)
and, by part (i), the possibility
∂(
r
)<∂(
u
)
cannot occur. Therefore,
r
=
0 and
u
is a unit, for
1
=
qu
.
3.70
Let
α
=
1
2
(
1
+
√
−
19
)
, and let
R
=
Z
[
α
]
.
(i)
Prove that
N
:
R
×
→
N
,de
f
ned by
N
(
m
+
n
α)
=
m
2
−
mn
+
5
n
2
,
is multiplicative:
N
(
u
v)
=
N
(
u
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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