## An exposition on the Kronecker-Weber theorem

dc.contributor.author | Baggett, Jason A. | |

dc.date.accessioned | 2020-10-15T16:48:43Z | |

dc.date.available | 2020-10-15T16:48:43Z | |

dc.date.issued | 2011-05 | |

dc.identifier.uri | http://hdl.handle.net/11122/11349 | |

dc.description | Thesis (M.S.) University of Alaska Fairbanks, 2011 | en_US |

dc.description.abstract | The Kronecker-Weber Theorem is a, classification result from Algebraic Number Theory. Theorem (Kronecker-Weber). Every finite, abelian extension of Q is contained in a cyclotomic field. This result was originally proven by Leopold Kronecker in 1853. However, his proof had some gaps that were later filled by Heinrich Martin Weber in 1886 and David Hilbert in 1896. Hilbert's strategy for the proof eventually led to the creation of the field of mathematics called Class Field Theory, which is the study of finite, abelian extensions of arbitrary fields and is still an area of active research. Not only is the Kronecker-Weber Theorem surprising, its proof is truly amazing. The idea of the proof is that for a finite, Galois extension K of Q, there is a connection between the Galois group Gal(K/Q) and how primes of Z split in a certain subring R of K corresponding to Z in Q. When Gal(K/Q) is abelian, this connection is so stringent that the only possibility is that K is contained in a cyclotomic field. In this paper, we give an overview of field/Galois theory and what the Kronecker-Weber Theorem means. We also talk about the ring of integers R of K, how primes split in R, how splitting of primes is related to the Galois group Gal(K/Q), and finally give a proof of the Kronecker-Weber Theorem using these ideas. | en_US |

dc.description.tableofcontents | 1. Introduction -- 2. Field/Galois theory summary -- 2.1. Algebraic field extensions -- 2.2. Automorphisms and the Galois Group -- 2.3. Cyclotomic fields -- 2.4. Finite fields -- 2.5. The Galois Correspondence theorem -- 2.6. The discriminant -- 3. Rings of algebraic integers -- 3.1. Algebraic integers -- 3.2. The trace and norm -- 3.3. The discriminant -- 3.4. The Kronecker-Weber theorem for quadratic extensions -- 3.5. Dedekind domains -- 4. Splitting of primes -- 4.1. Introduction -- 4.2. Ramification indices and inertial degrees -- 4.3. Splitting of primes in normal extensions -- 4.4. Ramification and the discriminant -- 4.5. The different -- 5. Decomposition, inertia, and ramification groups -- 5.1. Introduction -- 5.2. The main result -- 5.3. Some consequences of the main result -- 5.4. Splitting of primes in cyclotomic fields -- 5.5. Ramification groups -- 5.6. Hilbert's formula -- 6. The Kronecker-Weber theorem -- 6.1. Introduction -- 6.2. Special case: the field K has prime power degree pm over Q and p E Z is the only ramified prime -- 6.3. Special case: The field K has prime power degree over Q -- 6.4. General case -- 6.5. Examples -- Bibliography. | en_US |

dc.language.iso | en_US | en_US |

dc.subject | number theory | en_US |

dc.subject | algebraic number theory | en_US |

dc.subject | class field theory | en_US |

dc.subject | cyclotomic fields | en_US |

dc.subject | rings of integers | en_US |

dc.subject | Galois theory | en_US |

dc.title | An exposition on the Kronecker-Weber theorem | en_US |

dc.type | Thesis | en_US |

dc.type.degree | ms | en_US |

dc.identifier.department | Department of Mathematics | en_US |

refterms.dateFOA | 2020-10-15T16:48:44Z |