Abstract/Details

## On the minimum discriminant of algebraic number fields

Albrecht, William George, Jr.
University of South Florida ProQuest Dissertations Publishing,  1994. 9504477.

### Abstract (summary)

Let ${\cal C}$ be any nonempty collection of finite extensions of Q, such that if $K = Q(\theta)\in {\cal C}$, then K is a normal extension of Q. Let $\Delta\sb{K}$ be the discriminant for K as an extension of Q. Define\eqalign{&\qquad\qquad\quad\quad d({\cal C}) = {\rm min}\{\vert\Delta\sb{K}\vert : K\in{\cal C}\}\cr&{\cal C}\sb{G} = \{K : K\ {\rm is\ a\ finite\ extension\ of}\ Q\ {\rm with}\ G(K,Q)\cong G\}}and$$d(G) = d({\cal C}\sb{G}).$$

One of the main objectives of our research is to find d(G) for all groups that satisfy $\vert G\vert\le 10$ and to identify for each case a field K such that $\vert\Delta\sb{K}\vert$ = d(G).

Our results will be derived based mainly on the known structure of the quadratic and cyclotomic fields, results derived for the biquadratic fields, tables of cubic and quintic fields of small discriminant, class field theory and character theory. Much research has been done on finding the minimum discriminant of algebraic number fields of given degree with a prescribed number of real and complex roots, but little has been done based on the group theoretical behavior of normal extensions.

### Indexing (details)

Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences; discriminant
Title
On the minimum discriminant of algebraic number fields
Author
Albrecht, William George, Jr.
Number of pages
144
Degree date
1994
School code
0206
Source
DAI-B 55/09, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-209-24275-8
Liang, Joseph J.
University/institution
University of South Florida
University location
United States -- Florida
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9504477
ProQuest document ID
304128977