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- W1974056070 abstract "Richard G. Swan proved in 1962 that trinomials x8k+xm+1∈F2[x] with 8k>m have an even number of irreducible factors, and so cannot be irreducible. In fact, he found the parity of the number of irreducible factors for any square-free trinomial in F2[x]. We prove a result that is similar in spirit. Namely, suppose n is odd and f(x)=xn+∑i∈Sxi+1∈F2[x], where S⊂{i:iodd,0<i<n/3}∪{i:i=n(mod4),0<i<n}. We show that if n=±1(mod8) then f has an odd number of irreducible factors, and if n=±3(mod8) then f has an even number of irreducible factors. This has an application to the problem of finding polynomial bases {1,α,…,αn-1} of F2n such that Tr(αi)=0 for all 1⩽i<n." @default.
- W1974056070 created "2016-06-24" @default.
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- W1974056070 date "2006-01-01" @default.
- W1974056070 modified "2023-10-16" @default.
- W1974056070 title "A Swan-like theorem" @default.
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- W1974056070 doi "https://doi.org/10.1016/j.ffa.2005.02.001" @default.
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