a finite extension of fields is an algebraic extension
In order to prove that is an algebraic extension, we need to show that any element is algebraic, i.e., there exists a non-zero polynomial such that .
for some .
Consider the following set of “vectors” in :
Note that the cardinality of is , one more than the dimension of the vector space. Therefore, the elements of must be linearly dependent over , otherwise the dimension of would be greater than . Hence, there exist , not all zero, such that
Thus, if we define
then and , as desired.
NOTE: The converse is not true. See the entry “algebraic extension” for details.
|Title||a finite extension of fields is an algebraic extension|
|Date of creation||2013-03-22 13:57:30|
|Last modified on||2013-03-22 13:57:30|
|Last modified by||alozano (2414)|