(72) |

(73) |

(2) -> factor(x^105 - 1) (2) 2 4 3 2 (x - 1)(x + x + 1)(x + x + x + x + 1) * 6 5 4 3 2 (x + x + x + x + x + x + 1) * 8 7 5 4 3 (x - x + x - x + x - x + 1) * 12 11 9 8 6 4 3 (x - x + x - x + x - x + x - x + 1) * 24 23 19 18 17 16 14 13 x - x + x - x + x - x + x - x 12 11 10 8 7 6 5 + x - x + x - x + x - x + x - x + 1 * 48 47 46 43 42 41 40 x + x + x - x - x - 2x - x 39 36 35 34 33 - x + x + x + x + x 32 31 28 26 24 22 20 + x + x - x - x - x - x - x 17 16 15 14 13 + x + x + x + x + x 12 9 8 7 6 5 2 + x - x - x - 2x - x - x + x + x + 1 Type: Factored Polynomial Integer

(75) |

(76) |

*We define the measure of the polynomial
by
*

(77) |

- .
- .

(79) |

Therefore

(80) |

Let

(81) |

We calculate

(82) |

where we made repeated use of Lemma 3.

(83) |

(84) |

(85) |

(87) |

where are the roots of and thus among the roots of . Observe that the coefficient of (in the term of degree )

- is a sum of products, where each product consists of factors, each of these factors being a root of .
- Hence each of these factors is bounded by .
- Moreover there are of these factors. Indeed, in order to build one of these factors, one needs to choose roots among the of .

(88) |

This implies

Now observe that the measures of and satisfy

Relations (89) and (90) together we Theorem 4 leads to

(91) |

The theorem is proved.

(92) |

(93) |

From Theorem 5 (that essentially relates the -norm of a polynomial with its measure) we have

(94) |

From the elementary properties of the measure of a polynomial given in Remark 10 we have

(95) |

By virtue of Landau's inequality (Theorem 4) we have

(96) |

Putting all these relations together leads to

(97) |

The corollary is proved.

(98) |

*
*

2008-01-07