Quadratic Forms and Ideal Theory of Quadratic Fields

2014 ◽  
pp. 75-93
Author(s):  
Tomoyoshi Ibukiyama ◽  
Masanobu Kaneko
Keyword(s):  
Quadratic Forms ◽  
Ideal Theory ◽  
Quadratic Fields

scholarly journals UNIVERSAL QUADRATIC FORMS AND ELEMENTS OF SMALL NORM IN REAL QUADRATIC FIELDS

2016 ◽  
Vol 94 (1) ◽  
pp. 7-14 ◽  
Author(s):  
VÍTĚZSLAV KALA
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Small Norm ◽  
Real Quadratic

For any positive integer $M$ we show that there are infinitely many real quadratic fields that do not admit $M$-ary universal quadratic forms (without any restriction on the parity of their cross coefficients).

scholarly journals Continued fractions and class number two

2001 ◽  
Vol 27 (9) ◽  
pp. 565-571
Author(s):  
Richard A. Mollin
Keyword(s):  
Continued Fractions ◽  
Class Number ◽  
Quadratic Field ◽  
Ideal Theory ◽  
Quadratic Polynomial ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
New Class ◽  
Real Quadratic

We use the theory of continued fractions in conjunction with ideal theory (often called the infrastructure) in real quadratic fields to give new class number 2 criteria and link this to a canonical norm-induced quadratic polynomial. By doing so, this provides a real quadratic field analogue of the well-known result by Hendy (1974) for complex quadratic fields. We illustrate with several examples.

scholarly journals Number fields without n-ary universal quadratic forms

2015 ◽  
Vol 159 (2) ◽  
pp. 239-252 ◽  
Author(s):  
VALENTIN BLOMER ◽  
VÍTĚZSLAV KALA
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Number Fields ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

AbstractGiven any positive integer M, we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms with even cross coefficients in M variables.

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