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Kunze Linear Algebra
· Problem 4.2.8
Polynomials
Mathematics — Methods · polynomial-substitution-isomorphism
Problem
If
F
F
F
is a field and
h
h
h
is a polynomial over
F
F
F
of degree
≥
1
\geq 1
≥
1
, show that the mapping
f
→
f
(
h
)
f \to f(h)
f
→
f
(
h
)
is a one-one linear transformation of
F
[
x
]
F[x]
F
[
x
]
into
F
[
x
]
F[x]
F
[
x
]
. Show that this transformation is an isomorphism of
F
[
x
]
F[x]
F
[
x
]
onto
F
[
x
]
F[x]
F
[
x
]
if and only if
deg
h
=
1
\deg h = 1
de
g
h
=
1
.
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