Revision as of 01:02, 19 July 2015 edit David Eppstein (talk \| contribs) Autopatrolled, Administrators 235,886 edits →References: authorlink Harald Niederreiter (but someone else needs to take care of updating the footnotes for the change from 1st ed. 1983 to 2nd ed. 1997) ← Previous edit		Revision as of 02:41, 6 July 2016 edit undo Myasuda (talk \| contribs) Autopatrolled, Extended confirmed users, Pending changes reviewers 62,575 edits m →q-polynomials over Fq: sp Next edit →
Line 28: ==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~~factorization. Two important examples of this type of linearised polynomial are the Frobenius automorphism <math>x \mapsto x^q</math> and the trace function <math>\operatorname{Tr}(x) = \sum_{i=0}^{n-1} x^{q^i}</math>. In this special case 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:

Linearised polynomial: Difference between revisions