University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations May 2020 The Fundamental System of Units for Cubic Number Fields Janik Huth University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Other Mathematics Commons Recommended Citation Huth, Janik, "The Fundamental System of Units for Cubic Number Fields" (2020). Theses and Dissertations. 2385. https://dc.uwm.edu/etd/2385 This Thesis is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact
[email protected]. THE FUNDAMENTAL SYSTEM OF UNITS FOR CUBIC NUMBER FIELDS by Janik Huth A Thesis Submitted in Partial Fulllment of the Requirements for the Degree of Master of Science in Mathematics at The University of Wisconsin-Milwaukee May 2020 ABSTRACT THE FUNDAMENTAL SYSTEM OF UNITS FOR CUBIC NUMBER FIELDS by Janik Huth The University of Wisconsin-Milwaukee, 2020 Under the Supervision of Professor Allen D. Bell Let K be a number eld of degree n. An element α 2 K is called integral, if the minimal polynomial of α has integer coecients. The set of all integral elements of K is denoted by OK . We will prove several properties of this set, e.g. that OK is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group × , dened as the set of all invertible elements OK of OK .