<abstract xmlns="http://www.w3.org/1999/xhtml">Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, <italic>ℒ</italic>-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number <italic>h</italic>(<italic>d</italic>) formula in real quadratic fields claims that we have 
<disp-formula id="j_amns.2019.2.00033_ueq_001_w2aab3b7b1b1b6b1aab1c16b2b7Aa"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amns.2019.2.00033_ueq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>h</mi><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>.</mo><mi mathvariant="italic">log</mi><msub><mo>ε</mo><mi>d</mi></msub><mo>=</mo><msqrt><mrow><mo>Δ</mo><mo>.</mo></mrow></msqrt><mo>ℒ</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo> </mo><msub><mo>χ</mo><mi>d</mi></msub></mrow><mo>)</mo></mrow></mrow></math><tex-math>h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right)</tex-math></alternatives></disp-formula>
for positive <italic>d</italic> > 0 and the fundamental unit <italic>ɛ</italic><italic>d</italic> of 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amns.2019.2.00033_ieq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>ℚ</mo><mrow><mo>(</mo><mrow><msqrt><mi>d</mi></msqrt></mrow><mo>)</mo></mrow></mrow></math><tex-math>{\rm{\mathbb Q}}\left({\sqrt d} \right)</tex-math></alternatives></inline-formula>
. It is seen that discriminant, <italic>ℒ</italic>-function and fundamental unit <italic>ɛ</italic><italic>d</italic> are significant and necessary tools for determining the structure of real quadratic fields.The focus of this paper is to determine structure of some special real quadratic fields for <italic>d ></italic> 0 and <italic>d</italic> ≡ 2,3 (<italic>mod</italic>4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element <italic>w</italic><italic>d</italic>, fundamental unit <italic>ɛ</italic><italic>d</italic>, and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.</abstract>

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have 
h(d).logεd=Δ.ℒ(1, χd)h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right)
for positive d > 0 and the fundamental unit ɛd of 
ℚ(d){\rm{\mathbb Q}}\left({\sqrt d} \right)
. It is seen that discriminant, ℒ-function and fundamental unit ɛd are significant and necessary tools for determining the structure of real quadratic fields.The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element wd, fundamental unit ɛd, and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields"}

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and...

d	𝓁(d)	w_d	ɛ_d
123	2	$[11; \bar{11, 22}]$ \left[ {11;\overline {11,\;22}} \right]	$122 + 11 \sqrt{123}$ 122 + 11\sqrt {123}
465247	4	$[682; \bar{11, 11, 11, 1364}]$ \left[ {682;\overline {11,11,\;11,\;1364}} \right]	$922868 + 1353 \sqrt{465247}$ 922868 + 1353\sqrt {465247}
56371418	5	$[7508; \bar{11, 11, 11, 11, 15016}]$ \left[ {7508;\overline {11,11,\;11,11,\;15016}} \right]	$112658893 + 15005 \sqrt{56371418}$ 112658893 + 15005\sqrt {56371418}
104722907208735	8	$[10233421; \bar{11, 11, \dots, 11, 20466842}]$ \left[ {10233421;\overline {11,11,\; \ldots,11,\;20466842}} \right]	$209445700004344 + 20466801 \sqrt{104722907208735}$ 209445700004344 + 20466801\sqrt {104722907208735}

A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

Published Online: Oct 16, 2019

Page range: 495 - 498

Received: Mar 11, 2019

Accepted: Apr 02, 2019

DOI: https://doi.org/10.2478/amns.2019.2.00033

Keywords
Quadratic Fields, Continued Fraction Expansion, Fundamental Unit, Special Integer Sequences

© 2020 Özen Özer, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Numerical Results for Corollary 2.2.

Numerical Results for Corollary 2.1.

A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

Published Online: Oct 16, 2019

Page range: 495 - 498

Received: Mar 11, 2019

Accepted: Apr 02, 2019

DOI: https://doi.org/10.2478/amns.2019.2.00033

KeywordsQuadratic Fields, Continued Fraction Expansion, Fundamental Unit, Special Integer Sequences

© 2020 Özen Özer, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Numerical Results for Corollary 2.2.

Numerical Results for Corollary 2.1.

Keywords
Quadratic Fields, Continued Fraction Expansion, Fundamental Unit, Special Integer Sequences