existence of the minimal polynomial
We start by defining the following map:
Note that this map is clearly a ring homomorphism. For all :
Thus, the kernel of is an ideal of :
Note that the kernel is a non-zero ideal. This fact relies on the fact that is a finite extension of fields, and therefore it is an algebraic extension, so every element of is a root of a non-zero polynomial with coefficients in , this is, .
Let be the leading coefficient of . We define , so that the leading coefficient of is . Also note that by the previous remark, is the unique generator of which is monic.
By construction, , since belongs to the kernel of , so it satisfies .
Finally, if is any polynomial such that , then . Since generates this ideal, we know that must divide (this is property ).
For the uniqueness, note that any polynomial satisfying and must be a generator of , and, as we pointed out, there is a unique monic generator, namely .
|Title||existence of the minimal polynomial|
|Date of creation||2013-03-22 13:57:24|
|Last modified on||2013-03-22 13:57:24|
|Last modified by||alozano (2414)|