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Universal quadratic forms and indecomposables over biquadratic fields
Author(s) -
Čech Martin,
Lachman Dominik,
Svoboda Josef,
Tinková Magdaléna,
Zemková Kristýna
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800109
Subject(s) - indecomposable module , mathematics , quadratic equation , algebraic number , quadratic field , algebraic number field , field (mathematics) , pure mathematics , discrete mathematics , algebra over a field , mathematical analysis , quadratic function , geometry
The aim of this article is to study (additively) indecomposable algebraic integers O K of biquadratic number fields K and universal totally positive quadratic forms with coefficients in O K . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field K . Furthermore, estimates are proven which enable algorithmization of the method of escalation over K . These are used to prove, over two particular biquadratic number fields Q ( 2 , 3 ) and Q ( 6 , 19 ) , a lower bound on the number of variables of a universal quadratic forms.