Linearised polynomial

In mathematics, a linearised polynomial is a polynomial over a finite field of order q for which the exponents of all the consituent monomials are powers of q.

We write a typical example as

$L(x)=\sum _{i=0}^{n}a_{i}x^{q^{i}}\ .$

The map x → L(x) is a linear map on any field containing F_q and the set of roots of L is an F_q-vector space. Conversely, if U is any F_q-linear subspace of some finite field containg F_q, then the polynomial that vanishes exactly on U is a linearised polynomial.

Properties

• The set of linearised polynomials over a given field is closed under addition and composition of polynomials.

References

• Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069.