Linearised polynomial

In mathematics, a linearised polynomial (or "q"- polynomial) is a polynomial for which the exponents of all the constituent monomials are powers of "q" and the coefficients come from some extension field of the finite field of order "q".
We write a typical example as
This special class of polynomials is important from both a theoretical and an applications viewpoint. The highly structured nature of their roots makes these roots easy to determine.
Symbolic multiplication.
In general, the product of two linearised polynomials will not be a linearized polynomial, but since the composition of two linearised polynomials results in a linearised polynomial, composition may be used as a replacement for multiplication and, for this reason, composition is often called symbolic multiplication in this setting. Notationally, if "L"1("x") and "L"2("x") are linearised polynomials we define
when this point of view is being taken. 
Associated polynomials.
The polynomials L("x") and
are "q - associates" (note: the exponents ""q""i" " of "L"("x") have been replaced by "i" in "l"("x")). More specifically, "l(x}" is called the "conventional q-associate" of "L(x)", and "L(x)" is the "linearised q-associate" of "l(x)".
q-polynomials over F"q".
Linearised polynomials with coefficients in F"q" have additional properties which make it possible to define symbolic division, symbolic reducibility and symbolic factorization. Two important examples of this type of linearised polynomial are the Frobenius automorphism formula_7 and the trace function formula_8.
In this special case it can be shown that, as an operation, symbolic multiplication is commutative, associative and distributes over ordinary addition. Also, in this special case, we can define the operation of symbolic division. If "L"("x") and "L"1("x") are linearised polynomials over F"q", we say that "L"1("x") "symbolically divides" "L"("x") if there exists a linearised polynomial "L"2("x") over F"q" for which:
If "L"1("x") and "L"2("x") are linearised polynomials over F"q" with conventional q-associates "l"1("x") and "l"2("x") respectively, then "L"1("x") symbolically divides "L"2("x") if and only if "l"1("x") divides "l"2("x"). Furthermore, 
"L"1("x") divides "L"2("x") in the ordinary sense in this case.
A linearised polynomial "L"("x") over F"q" of degree > 1 is "symbolically irreducible" over F"q" if the only symbolic decompositions
with "L""i" over F"q" are those for which one of the factors has degree 1. Note that a symbolically irreducible polynomial is always reducible in the ordinary sense since any linearised polynomial of degree > 1 has the nontrivial factor "x". A linearised polynomial "L"("x") over F"q" is symbolically irreducible if and only if its conventional "q"-associate "l"("x") is irreducible over F"q".
Every "q"-polynomial "L"("x") over F"q" of degree > 1 has a "symbolic factorization" into symbolically irreducible polynomials over F"q" and this factorization is essentially unique (up to rearranging factors and multiplying by nonzero elements of F"q".)
For example, consider the 2-polynomial "L"("x") = "x"16 + "x"8 + "x"2 + "x" over F2 and its conventional 2-associate "l"("x") = "x"4 + "x"3 + "x" + 1. The factorization into irreducibles of "l"("x") = ("x"2 + "x" + 1)("x" + 1)2 in F2["x"], gives the symbolic factorization 
Affine polynomials.
Let "L" be a linearised polynomial over formula_2. A polynomial of the form formula_13 is an "affine polynomial" over formula_2.
Theorem: If "A" is a nonzero affine polynomial over formula_2 with all of its roots lying in the field formula_3 an extension field of formula_2, then each root of "A" has the same multiplicity, which is either 1, or a positive power of "q".