scholarly journals On a conjecture about class numbers of totally positive quadratic forms in totally real algebraic number fields

1979 ◽  
Vol 11 (2) ◽  
pp. 188-196 ◽  
Author(s):  
Horst Pfeuffer
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Number Fields ◽  
Class Numbers ◽  
Algebraic Number Fields ◽  
Totally Positive ◽  
Totally Real

scholarly journals There are no universal ternary quadratic forms over biquadratic fields

2020 ◽  
Vol 63 (3) ◽  
pp. 861-912 ◽  
Author(s):  
Jakub Krásenský ◽  
Magdaléna Tinková ◽  
Kristýna Zemková
Keyword(s):  
Quadratic Forms ◽  
Number Fields ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Mild Conditions ◽  
Positive Unit ◽  
Positive Elements ◽  
Totally Positive ◽  
Biquadratic Fields ◽  
Totally Real

AbstractWe study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

A note on class numbers of algebraic number fields

2001 ◽  
pp. 372-373
Author(s):  
Ichiro Satake ◽  
Genjiro Fujisaki ◽  
Kazuya Kato ◽  
Masato Kurihara ◽  
Shoichi Nakajima
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Class Numbers ◽  
Algebraic Number Fields
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