special reducible polynomials over a field with positive characteristic

Let k be an arbitrary field such that char(k)=p>0. We will assume that 0.

PropositionPlanetmathPlanetmath. Let m. Then for any ak the polynomial W(X)=Xpm-a is reducible if and only if there exist ck and n such that cpn=a. Moreover the factorization of W(X) is given by the formulaMathworldPlanetmathPlanetmath


where n is a maximal natural numberMathworldPlanetmath such that 0nm and a=cpn for some ck.

Proof. “” Assume that a=cpn for some ck and n. It is well known that if char(k)=p>0 and t then for any x,yk we have (x+y)pt=xpt+ypt. Therefore


Note that pm>deg(V(X))=pm-1>0 and therefore W(X) is reducible.

” Assume that W(X) is reducible. Therefore there exist V(X),U(X)k[X] such that W(X)=V(X)U(X) and both deg(V(X))>0 and deg(U(X))>0.

Recall that there exists an algebraically closed field k¯ such that k is a subfield of k¯ (generally it is true for any field). Therefore there exists c0k¯ such that c0pm=a and thus we have:


in k¯[X]. Now V(X)U(X)=W(X)=(X-c0)pm and since k¯[X] is a unique factorization domainMathworldPlanetmath then for n=deg(V(X))>0 we have:


But V(X)k[X] (the factorization was assumed to be over k) and therefore c0nk. It is easy to see that since c0nk and c0pmk then c0gcd(n,pm)k, but gcd(n,pm)=ps for some s. Thus if we put c=c0ps we gain that cpm-s=a. But m>s (since n<pm because we assumed that both deg(V(X))>0 and deg(U(X))>0), which completesPlanetmathPlanetmathPlanetmathPlanetmath the proof of the first part.

Now let n be a maximal natural number such that nm and a=cpn for some ck. Then we have


Note that the polynomial Xpm-n-c is irreduciblePlanetmathPlanetmath. Indeed, assume that Xpm-n-c is reducible. Then (due to first part of the proposition) c=upk for some k and uk. But then a=(upk)pn=upn+k. ContradictionMathworldPlanetmathPlanetmath, since n+k>n and n was assumed to be maximal.

Title special reducible polynomials over a field with positive characteristic
Canonical name SpecialReduciblePolynomialsOverAFieldWithPositiveCharacteristic
Date of creation 2013-03-22 18:31:05
Last modified on 2013-03-22 18:31:05
Owner joking (16130)
Last modified by joking (16130)
Numerical id 10
Author joking (16130)
Entry type Theorem
Classification msc 13F07