In: Advanced Math
Show that the map Q[X] → Q[X],
sum^{n}{i=0}(a_nX^i) → sum^{n}{i=0}(a_n(2X + 3)^i) , is an automorphism of Q[X],
Given function
,
We need to show that
is an automorphism. We first prove that it is a homomorphism.
Suppose
are given by
Then
and
Therefore,
is a ring homomorphism.
Thus, to prove that
is automorphism, it remains to show that it is bijective. We can
do this by showing that
has a two-sided inverse.
Consider the function
,
For any element
, given by
we have
and
Thus,
is a two-sided inverse of
. Hence,
is an automorphism.