Hello,

Let be the field
> q = 2
> K.<t> = GF(q^n)
> and the Polynomial Ring
> PR = PolynomialRing(K,"X")
> Let be a random monomial of PR for example
> P = t*X^(q^a).
> Is there any method in sage to reduce X degree of polynomial P, such that 
> equivalent polynomial is t*X^(q^b) where b = mod(P.degree(X)-1,n)
>

Does the following do what you want?

sage: n = 5; q = 2^n
sage: K.<t> = FiniteField(q)
sage: PR.<X> = PolynomialRing(K)
sage: Q.<Xbar> = PR.quotient(X^q - X)
sage: t*Xbar^(2*q)
t*Xbar^2

Peter

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