## Polynomials with Special Regard to ReducibilityThis book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields. |

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### Contents

8 | |

Lacunary polynomials over an arbitrary field | 92 |

Polynomials over an algebraically closed field | 201 |

Polynomials over a finitely generated field | 263 |

Polynomials over a number field | 315 |

Polynomials over a Kroneckerian field | 390 |

Appendices | 481 |

540 | |

555 | |

### Common terms and phrases

algebraic integer arbitrary arithmetically dense assume binomial Chapter char condition congruence conjugate consider const contains contradiction Convention coprime Corollary cyclotomic polynomial defined Definition degree denote dimension equation exist polynomials extension L/K F e K[x factor finite extension follows Galois group given Hilbertian holds implies inductive assumption inequality infer irreducible over k irreducible polynomial Kroneckerian field lattice Laurent polynomial leading coefficient Lemma 13 Lemma 9 Let F linear linearly independent log log Mahler measure monic monic polynomial monomials Moreover multiplicity non-zero coefficients number field number of terms obtain Omod polynomial F positive integer prime divisor prime ideals Proof of Theorem proved rank rational functions reducible overk residue class respect right hand side root of unity Schinzel Section self-inversive solvable subgroup subset Theorem 19 Theorem 60 variables zero in K(t

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Page 1 - prime if it is not the composition of two polynomials of lower degree and proved the two main results: (i) In every representation of a polynomial as the composition of prime polynomials the number of factors is the same and their degrees coincide up to a permutation. (ii) If

Page 11 - ord ¿, a is the highest power to which a prime element p of a unique factorization domain or a prime ideal p of a Dedekind domain divides an element a of this domain.