special reducible polynomials over a field with positive characteristic
Let be an arbitrary field such that . We will assume that .
where is a maximal natural number such that and for some .
Proof. “” Assume that for some and . It is well known that if and then for any we have . Therefore
Note that and therefore is reducible.
“” Assume that is reducible. Therefore there exist such that and both and .
Recall that there exists an algebraically closed field such that is a subfield of (generally it is true for any field). Therefore there exists such that and thus we have:
in . Now and since is a unique factorization domain then for we have:
But (the factorization was assumed to be over ) and therefore . It is easy to see that since and then , but for some . Thus if we put we gain that . But (since because we assumed that both and ), which completes the proof of the first part.
Now let be a maximal natural number such that and for some . Then we have
|Title||special reducible polynomials over a field with positive characteristic|
|Date of creation||2013-03-22 18:31:05|
|Last modified on||2013-03-22 18:31:05|
|Last modified by||joking (16130)|