276
6. RINGS
Example 6.2.17.
The kernel of the ring homomorphism
KOExŁ
!
K
given
by
p
7!
p.a/
is the set of all polynomials
p
having
a
as a root.
Example 6.2.18.
The kernel of the ring homomorphism
C.
R
/
!
C.S/
given by
f
7!
f
j
S
is the set of all continuous functions whose restriction
to
S
is zero.
Example 6.2.19.
Let
K
be a field.
Define a map
'
from
KOExŁ
to
Fun
.K; K/
, the ring of
K
–valued functions on
K
by
'.p/.a/
D
p.a/
.
(That is,
'.p/
is the polynomial function on
K
corresponding to the poly
nomial
p
.) Then
'
is a ring homomorphism. The homomorphism property
of
'
follows from the homomorphism property of
p
7!
p.a/
for
a
2
K
.
Thus
'.p
C
q/.a/
D
.p
C
q/.a/
D
p.a/
C
q.a/
D
'.p/.a/
C
'.q/.a/
D
.'.p/
C
'.q//.a/
, and similarly for multiplication.
The kernel of
'
is the set of polynomials
p
such that
p.a/
D
0
for all
a
2
K
. If
K
is infinite, then the kernel is
f
0
g
, since no nonzero polynomial
with coefficients in a field has infinitely many roots.
If
K
is finite, then
'
is
never
injective.
That is, there always exist
nonzero polynomials
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Ring, Homomorphism, ring homomorphism

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