89
Now so that the product lies in
F
4
. It also follows from this for
mula that multiplication in
F
4
is commutative. All the other ax
ioms of a commutative ring hold in
F
4
because they hold in the full
matrix ring(for the cogniscenti, every subring of a (not necessarily
commutative) ring is a ring).
(ii)
Prove that
F
4
is a
f
eld having exactly 4 elements.
Solution.
If a matrix
A
=
£
ab
ba
+
b
±
in
I
4
is nonzero, then
a
±=
0or
b
±=
0. Thus,
det
(
A
)
=
a
±=
0i
f
b
=
0
;
b
±=
f
a
=
0
;
1i
f
a
=
1
=
b
.
Thus, if
A
±=
0, then det
(
A
)
=
1, and so the matrix
A
−
1
exists.
As usual,
A
−
1
=
£
a
+
bb
±
, and so it is only a question of whether
this matrix lies in
I
4
; that is, is the 2
,
2 entry the sum of the entries
in row 1? The answer is yes, for
(
a
+
b
)
+
b
=
a
.
(iii)
Show that
I
4
is not a
f
eld.
Solution.
The commutative ring
I
4
is not even a domain, for
[
2
] ±=
[
0
]
and
[
2
][
2
]=[
4
0
]
.
3.20
Prove that every domain
R
with a
f
nite number of elements must be a
f
eld.
Solution.
Let
R
×
denote the set of nonzero elements of
R
. The cancella
tion law can be restated: for each
r
∈
R
×
, the function
µ
r
:
R
×
→
R
×
,
de
f
ned by
µ
r
:
x
7→
rx
, is an injection
R
×
→
R
×
. Since
R
×
is
f
nite,
Exercise 2.13 shows that every
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 Fall '11
 KeithCornell
 Addition, Ring, Ring theory

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