Linearised polynomial
Properties
Properties
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* The map  is a [linear map](/wiki/Linear_map "Linear map") over any [field](/wiki/Field_%28mathematics%29 "Field (mathematics)") containing **F***q*.
* The [set](/wiki/Set_%28mathematics%29 "Set (mathematics)") of roots of *L* is an **F***q*\-[vector space](/wiki/Vector_space "Vector space") and is closed under the *q*\-[Frobenius map](/wiki/Frobenius_map "Frobenius map").
* Conversely, if *U* is any **F***q*\-[linear subspace](/wiki/Linear_subspace "Linear subspace") of some finite field containing **F***q*, then the polynomial that vanishes exactly on *U* is a linearised polynomial.
* The set of linearised polynomials over a given field is closed under addition and [composition](/wiki/Function_composition "Function composition") of polynomials.
* If *L* is a nonzero linearised polynomial over F\_{q^n} with all of its roots lying in the field F\_{q^s} an extension field of F\_{q^n}, then each root of *L* has the same multiplicity, which is either 1, or a positive power of *q*.

Is the polynomial that vanishes exactly on U unique?