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Hi am trying to find as many irreducible polynomials in $\Bbb F_{2}[x]$ as possible and their generating cyclotomic cosets. So far I have found 3 - a number am quite pleased with; $1 + x + x^2 + x^3 + x^4, 1 + x + x^4 and 1 + x^3 + x^4$. I think these will be enough for me to get the concept, now i want to find the cyclotomic cosets that generate these polynomial. I know that cyclotomic cosets modulo $m$ are the sets of the form $C(j,m)=\{k\in\Bbb{Z}_{m}\mid k\equiv 2^i j\pmod{m}\ \text{for some natural number $i$}\}$. But how am i to decide which ones generate which of the above polynomial? Also, do all calculations of the cosets in this case have to be made $(mod 2)$? thanks.

guthik
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  • Look into http://www.ece.umd.edu/~abarg/626-2009/notes2005/lecture12.pdf There is a very instructive example on page 2. I could verify it easily with Macaulay 2 (there is a Macaulay 2 online calculator available too on the Macaulay 2 homepage). – Jürgen Böhm May 11 '15 at 00:22
  • oh, actually this is awesome. it is all i need. can you convert this to an answer so that i can accept it? – guthik May 11 '15 at 17:56

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Look into ece.umd.edu/~abarg/626-2009/notes2005/lecture12.pdf There is a very instructive example on page 2. I could verify it easily with Macaulay 2 (there is a Macaulay 2 online calculator available too on the Macaulay 2 homepage

Jürgen Böhm
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