Math 3124
Tuesday, November 8
November 8, Ungraded Homework
Problem 24.5 on page 124.
Prove that
Z
[
√
2
]
(that is
{
a
+
b
√
2

a
,
b
∈
Z
}
) is a ring.
We have the two operations of addition and multiplication as usual, so associativity and
distributivity are OK. The only thing that needs to be checked is that we have closure under
addition, subtraction (inverse for addition) and multiplication. Let
a
+
b
√
2,
c
+
d
√
2
∈
Z
[
√
2
]
(equivalently,
a
,
b
,
c
,
d
∈
Z
).
(
a
+
b
√
2
)+(
c
+
d
√
2
) = (
a
+
c
)+(
b
+
d
)
√
2
∈
Z
[
√
2
]
because
(
a
+
c
)
,
(
b
+
d
)
∈
Z
.

(
a
+
b
√
2
) =

a

b
√
2
∈
Z
[
√
2
]
because

a
,

b
∈
Z
.
(
a
+
b
√
2
)(
c
+
d
√
2
) = (
ac
+
2
bd
)+(
ad
+
bc
)
√
2
∈
Z
[
√
2
]
because
(
ac
+
2
bd
)
,
(
ad
+
bc
)
∈
Z
.
Problem 24.15 on page 125.
Let
E
denote the set of even integers. Prove that with the
usual addition, and with multiplication deﬁned by
m
*
n
=
mn
/
2, then
E
is a ring. Does it
have a unity?
Addition makes
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 Fall '08
 PARRY
 Algebra, Addition, Multiplication, Ring, ad + bc

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