Thesis (Selection of subject)Thesis (Selection of subject)(version: 285)
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Universal quadratic forms over number fields
Thesis title in Czech: Univerzální kvadratické formy nad číselnými tělesy
Thesis title in English: Universal quadratic forms over number fields
Key words: univerzální kvadratická forma, číselné těleso, bikvadratické, řetězový zlomek, aditivně nerozložitelný prvek
English key words: universal quadratic form, number field, biquadratic, continued fraction, additively indecomposable element
Academic year of topic announcement: 2015/2016
Type of assignment: Bachelor's thesis
Thesis language: angličtina
Department: Department of Algebra (32-KA)
Supervisor: Mgr. Vítězslav Kala, Ph.D.
Author: hidden - assigned and confirmed by the Study Dept.
Date of registration: 15.10.2015
Date of assignment: 17.10.2015
Confirmed by Study dept. on: 24.05.2016
Date and time of defence: 17.06.2016 00:00
Date of electronic submission:26.05.2016
Date of submission of printed version:27.05.2016
Date of proceeded defence: 17.06.2016
Reviewers: Ing. Tomáš Hejda, Ph.D.
A quadratic form over a totally real number field K is universal if it is totally positive and represents all totally positive elements of the ring of integers O_K. Recently, the arity of universal forms over a real quadratic field Q(sqrt D) has been connected [1, 3] to the existence of certain (additively indecomposable) elements [2] via the continued fraction expansion of sqrt D.
The goal of the thesis is to work out in detail the related theory (including some relevant results from algebraic number theory) and to get acquainted with these results. The student will illustrate the results on examples of specific fields and study similar problems in the setting of number fields of higher degree, especially biquadratic ones.
[1] V. Blomer, V. Kala, Number fields without universal n-ary quadratic forms, Math. Proc. Cambridge Philos. Soc. 159 (2015), 239-252.
[2] S. W. Jang, B. M. Kim, A refinement of the Dress–Scharlau theorem, J. Number Theory 158 (2016), 234-243.
[3] V. Kala, Universal quadratic forms and elements of small norm in real quadratic fields, preprint.
[4] J. S. Milne: Algebraic Number Theory,
[5] M. R. Murty, J. Esmonde: Problems in Algebraic Number Theory, GTM 190.
[6] S. Lang: Algebraic Number Theory, GTM 110.
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