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* The set of linearised polynomials over a given field is closed under addition and composition of polynomials.
==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''<sub>1</sub>(''x'') and ''L''<sub>2</sub>(''x'') are linearised polynomials we define
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when this point of view is being taken.
In the special case that the linearised polynomials are defined over '''F'''<sub>''q''</sub>, it can be shown that, as an [[Operation (mathematics)|operation]], symbolic multiplication is [[Commutative property|commutative]], [[associative]] and [[Distributive property|distributes]] over ordinary addition.<ref>{{harvnb|Lidl|Niederreiter|1983|loc=pg. 115 (first edition)}}</ref> Also, in this special case, we can define the operation of '''symbolic division'''. If ''L''(''x'') and ''L''<sub>1</sub>(''x'') are linearised polynomials over '''F'''<sub>''q''</sub>, we say that ''L''<sub>1</sub>(''x'') ''symbolically divides'' ''L''(''x'') if there exists a linearised polynomial ''L''<sub>2</sub>(''x'') over '''F'''<sub>''q''</sub> for which:▼
:<math>L(x) = L_1(x) \otimes L_2(x).</math> ▼
==Associated polynomials==
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are ''q - associates'' (note: the exponents "''q''<sup>''i''</sup> " of ''L''(''x'') have been replaced by "''i''" in ''l''(''x'')).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'''<sub>''q''</sub>==
Linearised polynomials with coefficients in '''F'''<sub>''q''</sub> have additional properties which make it possible to define symbolic division, symbolic reducibility and symbolic factorizaton.
▲In
▲:<math>L(x) = L_1(x) \otimes L_2(x).</math>
If ''L''<sub>1</sub>(''x'') and ''L''<sub>2</sub>(''x'') are linearised polynomials over '''F'''<sub>''q''</sub> with conventional q-associates ''l''<sub>1</sub>(''x'') and ''l''<sub>2</sub>(''x'') respectively, then ''L''<sub>1</sub>(''x'') symbolically divides ''L''<sub>2</sub>(''x'') if and only if ''l''<sub>1</sub>(''x'') divides ''l''<sub>2</sub>(''x'').<ref>{{harvnb|Lidl|Niederreiter|1983|loc=pg. 115 (first edition) Corollary 3.60}}</ref> Furthermore,
''L''<sub>1</sub>(''x'') divides ''L''<sub>2</sub>(''x'') in the ordinary sense in this case.<ref>{{harvnb|Lidl|Neiderreiter|1983|loc=pg. 116 (first edition) Theorem 3.62}}</ref>
A linearised polynomial ''L''(''x'') over '''F'''<sub>''q''</sub> of degree > 1 is ''symbolically irreducible'' over '''F'''<sub>''q''</sub> if the only symbolic decompositions
::<math>L(x) = L_1(x) \otimes L_2(x),</math>
with ''L''<sub>''i''</sub> over '''F'''<sub>''q''</sub> are those for which one of the factors has degree 1. Note that a symbolically irreducible polynomial is always [[Reducible expression|reducible]] in the ordinary sense since any linearised polynomial of degree > 1 has the nontrivial factor ''x''. A linearised polynomial ''L''(''x'') over '''F'''<sub>''q''</sub> is symbolically irreducible if and only if its conventional ''q''-associate ''l''(''x'') is irreducible over '''F'''<sub>''q''</sub>.
Every ''q''-polynomial ''L''(''x'') over '''F'''<sub>''q''</sub> of degree > 1 has a ''symbolic factorization'' into symbolically irreducible polynomials over '''F'''<sub>''q''</sub> and this factorization is essentially unique (up to rearranging factors and multiplying by nonzero elements of '''F'''<sub>''q''</sub>.)
For example,<ref>{{harvnb|Lidl|Neiderreiter|1983|loc=pg. 117 (first edition) Example 3.64}}</ref> consider the 2-polynomial ''L''(''x'') = ''x''<sup>16</sup> + ''x''<sup>8</sup> + ''x''<sup>2</sup> + ''x'' over '''F'''<sub>2</sub> and its conventional 2-associate ''l''(''x'') = ''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x'' + 1. The factorization into irreducibles of ''l''(''x'') = (''x''<sup>2</sup> + ''x'' + 1)(''x'' + 1)<sup>2</sup> in '''F'''<sub>2</sub>[''x''], gives the symbolic factorization
::<math>L(x) = (x^4 + x^2 + x) \otimes (x^2 + x) \otimes (x^2 + x).</math>
==Notes==
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