Quadratic Fields
An Algorithm for Explicit Form of Fundamental Units of Certain Real Quadratic Fields and Period Eight
... real quadratic number fields. This new algorithm for such quadratic fields is first in the literature and it gives us a more practical way to calculate the fundamental ...the quadratic ...
14
Class Number Formula for Certain Imaginary Quadratic Fields
... two quadratic fields Q （ 4p ） and Q （ - p ） as sub-fields of Q （ i , 4p ） , and interpret ...imaginary quadratic field Q （ - 4p ） in the case ...
6
Euler systems for modular forms over imaginary quadratic fields
... A third approach to the study of Selmer groups for modular forms over imaginary quadratic fields is to be found in the work of Skinner and Urban [SU14]. Their approach relies on establishing a lower bound ...
43
Explicit Form of Fundamental Units of Certain Real Quadratic Fields
... real quadratic fields Q( p d ) where d is congruent to 1 modulo 4 and the period k d in the continued fraction expansion of the quadratic irrational number ω d in Q( p d ) is equal to 3 and 4, 5 was ...
10
Representation of Algebraic Integers as Sum of Units over the Real Quadratic Fields
... imaginary quadratic fields 𝑄(√𝑑), 𝑑 < 0, 𝑑 = −1, −2, −3 as sum of units of this field of certain repetition 𝑡, 𝑡 ≥ ...imaginary fields in 𝑊 𝑡 ...
5
Torsion of elliptic curves over real quadratic fields of smallest discriminant
... −1) only. In 2010, Najman [9, 10] took a different approach by fixing a quadratic extension K of Q and then looking for possible torsion structure over K. A precise list of torsion subgroups for the ...
9
On the real quadratic fields with certain continued fraction expansions and fundamental units
... A quadratic field is defined as an algebraic number field Q( √ d) of degree two over Q the rational ...real quadratic field if d > ...many quadratic fields and all of them have class ...
12
On Lifting and Modularity of Reducible Residual Galois Representations Over Imaginary Quadratic Fields
... In this section we will state our main theorems (Theorems 8.1, 8.2 and 8.5) for the two-dimensional Galois representations over imaginary quadratic fields considered in section 7. In this case many of the ...
30
On elliptic curves of prime power conductor over imaginary quadratic fields with class number one
... imaginary quadratic fields of class number one to well-known bounds relating the conductor and discriminant of an elliptic curve over Q with prime power ...imaginary quadratic fields that do ...
34
Prime valued polynomials and class numbers of quadratic fields
... It is the purpose of this paper to give a survey of the relationship between the class number one problem for real quadratic fields and prime-producing quadratic polynomials; culminating[r] ...
11
On the calculation of regulators and class numbers of quadratic fields
... is that every equivalence class contains exactly one reduoed form. In the real quadratic case, this is not true any more; here every equivalence class contains a whole oyole of reduced f[r] ...
28
On the Mixed Littlewood Conjecture and continued fractions in quadratic fields
... xed quadratic irrationality and t belongs to the group of M -units O M × for a xed nite set of primes M ...a quadratic irrationality, an explicit subset S ′ of de Mathan's set S ...
11
p Capitulation over Number Fields with p Class Rank Two
... Remark 3.4. In 2012, Bembom investigated the 5-capitulation over complex quadratic fields K with 5-class group of type (5, 5) ([15] p. 129). However, his techniques were only able to distinguish between ...
14
Relationships Between Prior Experiences, Current Teaching Contexts, and Novice Teachers' Use of Concrete Representation for Mathematics Instruction
... It appears the log log philosophy holds true for the fake real quadratic case. As D varies with fixed p, the probability of failure is D 1 , which is expected since there is no reason why we believe D should not ...
92
Calculation of fundamental units in some types of quartic number fields : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University
... complex quadratic fields , and an algorithm by Arnara [1981] for calculating the class group and fundamental unit of a quadratic extension of a complex quadratic field for the case wheYe the ...
220
The Rabinowitsch Mollin Williams Theorem Revisited
... If I, J are O Δ -ideals, then equivalence of classes in C Δ is denoted by I ∼ J, and the class of I is denoted by I. The following is crucial to the interplay between ideals and continued fractions, known as the ...
14
Successive Approximation of p Class Towers
... currently the maximal proven finite length, in Theorems 4.5 and 4.6, is entirely due to our cooperation with M. R. Bush, initiated by our joint paper [36]. With Theorems 5.1 and 6.1, we have finally presented a new ...
26
Index p Abelianization Data of p Class Tower Groups
... complex quadratic fields with type ( ) 9, 3 in the range − 10 8 < < d 0 of discriminants, Again, we found exactly 5 errors among these IPADs which had been computed by PARI/GP [1] in ...
28
A mod p variant of the André–Oort conjecture
... (n ≥ 1), and consist of the pairs ( j( E), j (E/ P)) with E a complex elliptic curve and P ∈ E of order n. In this case, the conjecture was proved in [1], and, conditionally on the generalised Riemann hypothesis (GRH) ...
7
Introductory remarks on complex multiplication
... Stark, H.M., Class numbers of complex quadratic fields, Modular Functions of One Variable I, Antwerp, 19P2.. Weber, H., E_l__l_iPtische Funktionen.[r] ...
16