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Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1988 Class Numbers and Units of Number Fields E With Elementary Abelian K(2)O(E). Ruth Ilse Berger Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Berger, Ruth Ilse, "Class Numbers and Units of Number Fields E With Elementary Abelian K(2)O(E)." (1988). LSU Historical Dissertations and Theses. 4483. https://digitalcommons.lsu.edu/gradschool_disstheses/4483 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS The most advanced technology has been used to photo­ graph and reproduce this manuscript from the microfilm master. UMI films the original text directly from the copy submitted. Thus, some dissertation copies are in typewriter face, while others may be from a computer printer. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyrighted material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are re­ produced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each oversize page is available as one exposure on a standard 35 mm slide or as a 17" x 23" black and white photographic print for an additional charge. Photographs included in the original manuscript have been reproduced xerographically in this copy. 35 mm slides or 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ■Accessing Uthe World’s Information M since 1938 I 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA Order Number 8819923 Class numbers and units of number fields E with elementary ab elia n K 2 (Oj?;) Berger, Ruth Ilse, Ph.D. The Louisiana State University and Agricultural and Mechanical Col., 1988 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 Class numbers and units of number fields E with elementary abelian K iiO s) A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Ruth Ilse Berger Vordiplom, Universitat des Saarlandes, 1981 M.S., Louisiana State University, 1985 May 1988 Acknowledgments I would like to thank my advisor Dr. J. Hurrelbrink for sug­ gesting the problem and for supervising my progress on it. I would especially like to thank Dr. P.E. Conner who suggested many of the results and who helped me find access to their proofs. I am very grateful to Dr. J.W. Hoffman who was always available to discuss problems and who spent a lot of time on reading the first version and pointing out several mistakes. Table of Contents A b s t r a c t ..................................... iv Introduction ............................................................................................................................... v Chapter 1: Approaching the problem .............................................................................1 1. Collection of facts about A ^ O f ) ......................................................................................2 2. Quadratic number fields ......................................................................................................7 3. Biquadratic dicyclic number fields ....................................................................................9 4. Biquadratic cyclic number fields .................................................................................... 11 Chapter 2: Number fields with property (*) ................................................................ 18 5. Setting up the tools ..............................................................................................................19 6. The exact hexagon ... ..........................................................................................................26 7. Characterizing number fields with property (*) ............................................................29 8. The existence of number fields with property (*) ........................................................34 Chapter 3: The complete picture .................................................................................... 40 9. Detailed properties ............................................................................................................... 40 10. Completions and generalized ideal class groups ......................................................44 11. Families of number fields with property (*) ..............................................................53 12. The m ain th e o re m ..............................................................................................................60 Chapter 4: Examples ........................................................................................................... 69 13. A pplication to F = Q ........................... 70 14. Biquadratic dicyclic number fields with property (*) ............................................73 15. The fields Q(v^\/2?) with q = 5 mod 8 ........................................................................74 16. Quadratic extensions of Q(\/l0) with property (*) .................................................78 Bibliography 83 Class numbers and units of number fields E with elementary abelian K ^ O e ). by Ruth I. Berger A b stract This is a contribution to the research that is going on in Algebraic Number Theory, relating classical questions on class numbers and units of a number field F to the structure of (O f ) > the Milnor K-group K 2 of the ring of integers. We are interested in number fields F where the 2-primary subgroup of O f ) is elementary abelian of rank r1(F), the number of real embeddings of F. In [C-jHi ] it is proven that the 2-primary subgroup of /^ ( O f ) °f the above type if and only if the number field has the following properties: a) the number field has exactly one dyadic prime, b) its S-dass number is odd and c.) it contains S-units with independent signs. Here, the set S consists of all dyadic and all infinite primes of the number field. The purpose of this paper is to examine the existence of number fields of the above type and to examine their properties with respect to the parity of their class number and the containment of units with independent signs. We will mostly restrict our attention to number fields that are totally real. For any given totally real number field F that satisfies the above properties we will prove that there exist infinitely many real quadratic extensions that also have the above properties. The main theorem will be a classification of these quadratic extensions of F into families that all share the same properties with respect to the parity of their class number and the containment of units with independent signs. Introduction Let us first rewiew the objects that classical Number Theory deals with. The field of rational numbers Q contains the ring of integers TL. The integers contain two kinds of elements that stand out: the units and the prime elements. The units are defined as the integers u with the following property: there exists an integer x such that ux — 1. The elements of TL that satisfy this property are 1 and —1. The prime elements of TL are defined as the integers p with the following property: if a, b are integers such that ab = p then either a or b must equal p, up to unit factors ±1. The units and the prime elements are the “building blocks” for all integers. Up to factors of ±1, every integer can be expressed uniquely as a product of powers of prime numbers. Instead of considering the field (Q, one can look more generally at a num ber field. These fields are defined as the finite algebraic extensions of Q. This means that a finite number of roots of polynomials with coefficients in <Q are adjoined to Q. A number field F shares many of the properties of Q. F also contains what are called integers. They are defined as the integral closure of the rational integers, i.e., the roots of monic polynomials whose coefficients are in TL. The integers of a number field F are denoted by Of - Among the integers there are elements called units. As before, they are defined as the integers u with the property that there exists an integer x such that ux = 1. Note th at x must then be a unit, too. The units form a multiplicative group. It is denoted by Op. In general, there are no integers that behave like the prime elements(numbers) of TL] there are no “smallest factors” of which all integers can be uniquely ex­ pressed as a product. Therefore the idea of integers and prime elements needs to be generalized to objects that reflect this property of being minimal factors. This is done by introducing the concept of ideals. They are subsets of the ring of integers that are obtained by taking a set of integers, called the generators of the ideal, and taking all possible finite sums of products of these elements with integers. If the generators are taken

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