ON THE NUMBERS REPRESENTED BY CUBIC POLYNOMIALS
ON THE NUMBERS REPRESENTED BY CUBIC POLYNOMIALS
- 주제(키워드) cubic polynomials
- 발행기관 아주대학교
- 지도교수 방승진
- 발행년도 2010
- 학위수여년월 2010. 8
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/ajou/000000010900
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
In many papers and books, we can find some properties about the numbers represented by quadratic polynomials, especially the norm in quadratic fields. But, it is not easy to see properties about the numbers represented by cubic polynomials in cubic fields. M. A. Mathews[7] found the fact that for some integer , there is infinitely many triples of integers for special cubic equatons. E. J. Barbeau[3] found solutions for special cases of the parameter c of special cubic equations. In this thesis, we study about the numbers represented by the cubic polynomial in cubic fields. First, we prove Lemma 3.3.4. Second, we Theorem 3.3.5. Third, we prove Theorem 3.3.6. Fourth, we prove Corollary 3.3.8,, 3.3.9, 3.3.10, and 3.3.11. Fifth, we find the fact that there is no integral solution of soem Diophantine equations, see Example 4.1.1, 4.1.2, 4.1.3, and 4.1.4.
more목차
1. Introduction 1
1.1 Quadratic equation 2
1.2 Cubic equation 7
2. Preliminaries 10
2.1 Quadratic fields 10
2.2 Cubic fields 18
3. The representable numbers by the cubic polynomials 25
3.1 Fundamental theorems 27
3.2 Mathews and Barbeau's theorems 32
3.3 The representable numbers by the cubic
polynomials 35
4. Applications 47
4.1 Diophantine equation 47
4.2 Suggestions 49
REFERENCES 51
