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factor cyclotomic polynomial explicitly where $Φ_{17} (X)$ over $\mathrm{GF}(2) $

$(x-1)^{m(17/1)} (x^{17}-1)^{m(17/17)}$

How can I complete my steps

Simon Fraser
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    Welcome to Mathematics Stack Exchange. Do you mean cyclotomic? Are you familiar with Wolfram Alpha? – J. W. Tanner Dec 30 '19 at 16:55
  • i mean next step of solution over GF(2) , i don't mean factor x^17-1 –  Dec 30 '19 at 17:04
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    ${ _{17} (X) over GF(2) }$ –  Dec 30 '19 at 17:08
  • Wolfram Alpha gives the answer to that too -- look carefully – J. W. Tanner Dec 30 '19 at 17:22
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    Using the piece of theory I outlined here you can show that your polynomial will have two factors of degree eight. This also follows from the cyclicity of the multiplicative groups of finite fields - you look for the smallest integer $m>0$ such that $17\mid 2^m-1$. Furthermore, you can figure out whether the two factors are each others reciprocals or self-reciprocal i.e. palindromic. That will give you a lot of information fo work with. – Jyrki Lahtonen Dec 30 '19 at 17:36
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    And if the educated guessing from the previous comment fails to give you a useful system of equations (in the unknown coefficients of the factors), you can apply a factorization algorithm on it. For an on site example of Berlekamp's algorithm look here. I don't know whether Mathematica/Wolfram Alpha uses Berlekamp or the somewhat faster but non-deterministic Cantor-Zassenhaus. – Jyrki Lahtonen Dec 30 '19 at 17:58
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    Last but not least. Welcome to the site! To make sure your questions don't attract negative attention please study our guide for new askers. – Jyrki Lahtonen Dec 30 '19 at 18:00
  • thank you for your advise –  Dec 30 '19 at 18:13

1 Answers1

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Using the Berlekamp algorithm over $\Bbb F_2$ we have $$ \Phi_{17}(X)=1+X+X^2+\cdots +X^{16}=(X^8 + X^7 + X^6+X^4 +X^2+X+1)(X^8 + X^5 + X^4+X^3+1). $$

Dietrich Burde
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