We shall talk about the algebraic extension . We shall assume that both
and
are algebraic over the field
.
Assume . Hence, the basis for the vector space
over
is
. Now say
. Then the basis of
over
is
.
It is known that the basis of over
is
(arranged as a matrix).
None of these can be dependant on each other (by definition). Also, they are in number.
Now let us first construct from
. Say
. Then the basis of
over
is
. Now let us assume
over
is
. The basis matrix of
over
will hence be
over
is
.
It is possible that not a single term in the two matrices are the same. However, .
Remember.