## Number Theory in Function FieldsElementary number theory is concerned with the arithmetic properties of the ring of integers, Z, and its field of fractions, the rational numbers, Q. Early on in the development of the subject it was noticed that Z has many properties in common with A = IF[T], the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold for Z have analogues of the ring A. This is indeed the case. The first four chapters of this book are devoted to illustrating this by presenting, for example, analogues of the little theorems of Fermat and Euler, Wilson's theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet's theorem on primes in an arithmetic progression. All these results have been known for a long time, but it is hard to locate any exposition of them outside of the original papers. Algebraic number theory arises from elementary number theory by con sidering finite algebraic extensions K of Q, which are called algebraic num ber fields, and investigating properties of the ring of algebraic integers OK C K, defined as the integral closure of Z in K. |

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### Contents

Primes Arithmetic Functions and the Zeta Function | 11 |

Dirichlet LSeries and Primes in an Arithmetic Progression | 33 |

Weil Differentials and the Canonical Class | 63 |

Extensions of Function Fields RiemannHurwitz | 77 |

Constant Field Extensions | 101 |

Galois Extensions Hecke and Artin LSeries | 115 |

Artins Primitive Root Conjecture | 149 |

The Behavior of the Class Group in Constant Field Extensions | 169 |

Cyclotomic Function Fields | 193 |

An Introduction | 219 |

The BrumerStark Conjecture | 257 |

The Class Number Formulas in Quadratic | 283 |

Average Value Theorems in Function Fields | 305 |

341 | |

355 | |

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### Common terms and phrases

abelian extension AK(D algebraic closure algebraic number assertion assume automorphism Brumer-Stark conjecture Chapter class group class number coefficients consider constant field extension Corollary cyclic cyclotomic d-th power define definition deg(m degK denote differential Dirichlet character Dirichlet density dividing divisor classes Drinfeld module effective divisors equal equation extension of function fact field extension field F finite field follows formula Galois extension Galois group genus geometric global function fields homomorphism integral closure irreducible polynomial isomorphism L-functions Lemma Let L/K LK(u lying maximal ideal monic irreducible monic polynomials non-trivial non-zero notation number fields number theory ordp(a pole polynomial of degree positive integer prime decomposition prime ideal primes of degree properties Proposition prove quadratic ramified Recall relatively prime result Riemann hypothesis ring root of unity set of primes splits completely square-free subgroup Suppose unramified vanishes vector space zeta function