| 1. |
- Browkin, J., et al.
(författare)
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On Sequences of Squares with Constant Second Differences
- 2006
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Ingår i: Canadian Mathematical Bulletin. ; 49:4, s. 481-491
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this paper is to study sequences of integers for which the second differences between their squares are constant. We show that there are infinitely many nontrivial monotone sextuples having this property and discuss some related problems.
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| 2. |
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| 3. |
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| 4. |
- Browkin, J., et al.
(författare)
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Separable free quadratic algebras over quadratic integers
- 2004
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Ingår i: Journal of Number Theory. - : Elsevier BV. - 0022-314X .- 1096-1658. ; 109:2, s. 379-389
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Tidskriftsartikel (refereegranskat)abstract
- The aim of the paper is to determine all free separable quadratic algebras over the rings of integers of quadratic fields in terms of the properties of the fundamental unit in the real case. The paper corrects some earlier published results on the subject. © 2004 Elsevier Inc. All rights reserved.
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| 5. |
- Brzezinski, Juliusz, 1939
(författare)
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Corrigendum to "On traces of the Brandt-Eichler matrices"
- 2017
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Ingår i: Journal de Theorie des Nombres de Bordeaux. - 1246-7405. ; 29:1, s. 321-325
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Tidskriftsartikel (refereegranskat)abstract
- Abstract. This is a correction to my paper [1]. It corrects a formula in Proposition 2.2. The corrected result says that the number ı(n, m) of principal left ideals with norm qm in the Eichler order of level n over a discrete valuation ring R with residue field of cardinality q is ı(n, m) = (m + 1)qm if m < n and ı(n, m) = (n + 1)qm + 2qm−1 + · · · + 2qⁿ + qn−1 when m ≥ n. The proof of the Proposition was not given in my paper (as “tedious but straightforward”). Unfortunately, some coefficients in the second case were erroneous. A complete proof follows below.
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| 6. |
- Brzezinski, Juliusz, 1939
(författare)
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Galois groups and number theory
- 2011
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Ingår i: Normat - Nordisk Matematisk Tidskrift. - 0801-3500. ; 59:3-4, s. 144-177
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Tidskriftsartikel (refereegranskat)abstract
- This is a survey article on algebraic number theory and its purpose is to show the great importance of Galois groups in different number theoretical problems. In particular, the role of Galois representations in connection with the proof of Fermat's Last Theorem is one of the topics.
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| 7. |
- Brzezinski, Juliusz, 1939
(författare)
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Galois Theory Through Exercises
- 2018
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- This textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises). In addition to covering standard material, the book explores topics related to classical problems such as Galois’ theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading. A valuable reference and a rich source of exercises with sample solutions, this book will be useful to both students and lecturers. Its original concept makes it particularly suitable for self-study.
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| 8. |
- Brzezinski, Juliusz, 1939
(författare)
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Hilberts Tionde Problem och Büchisekvenser : Hilbert's Tenth Problem and Büchi's Sequences
- 2012
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Ingår i: Nordisk Matematisk Tidskrift, Normat. - 0801-3500. ; 60:2, s. 52-69
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Tidskriftsartikel (refereegranskat)abstract
- The paper gives an overview of Büchi's square problem in the context of Hilbert's Tenth Problem. Several construction methods of integer sequences with constant second differences of squares are discussed and 5 new nontrivial septuplets of this type are presented.
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| 9. |
- Brzezinski, Juliusz, 1939
(författare)
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Hilberts Tionde Problem och Büchisekvenser
- 2012
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Ingår i: Nordisk Matematisk Tidskrift, Normat. - 0801-3500. ; 60:2, s. 52-69
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Tidskriftsartikel (refereegranskat)abstract
- The paper gives an overview of Büchi's square problem in the context of Hilbert's Tenth Problem. Several construction methods of integer sequences with constant second differences of squares are discussed and 5 new nontrivial septuplets of this type are presented.
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| 10. |
- Brzezinski, Juliusz, 1939, et al.
(författare)
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How to describe all cubic Galois extensions?
- 2011
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Ingår i: Normat-Nordisk Matematisk Tidskrift. - 0801-3500. ; 59:3-4, s. 117-130
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Tidskriftsartikel (refereegranskat)abstract
- All cubic Galois extensions of rational numbers are described by canonical minimal polynomials, which can be computed using a "sieving" algorithm.
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| 11. |
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| 12. |
- Brzezinski, Juliusz, 1939
(författare)
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On Exceptions in the Brauer-Kuroda Relations
- 2011
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Ingår i: Bulletin of the Polish Academy of Sciences Mathematics. - 0239-7269. ; 59:3, s. 207-214
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Tidskriftsartikel (refereegranskat)abstract
- Let F be a Galois extension of a number field k with Galois group G. The Brauer-Kuroda theorem gives an expression of the Dedekind zeta function of the field F as a product of the zeta functions of some of its subfields containing k, provided the group G is not exceptional. In this paper, we investigate the exceptional groups. In particular, we determine all nilpotent exceptional groups and give a sufficient condition for a group to be exceptional. We give many examples of nonnilpotent solvable and nonsolvable exceptional groups.
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| 13. |
- Brzezinski, Juliusz, 1939, et al.
(författare)
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On integral similitude matrices
- 2009
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Ingår i: Colloquium Mathematicum. - 0010-1354 .- 1730-6302. ; 115:1, s. 1-12
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Tidskriftsartikel (refereegranskat)abstract
- We study integral similitude 3 x 3-matrices and those positive integers which occur as products of their row elements, when matrices are symmetric with the same numbers in each row. It appears that integers for which nontrivial matrices of this type exist define elliptic curves of nonzero rank and are closely related to generalized cubic Fermat equations.
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| 14. |
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| 15. |
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| 16. |
- Brzezinski, Juliusz, 1939, et al.
(författare)
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Smooth lattices over quadratic integers
- 2008
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Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 1432-1823 .- 0025-5874. ; 258:1, s. 161-184
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Tidskriftsartikel (refereegranskat)abstract
- We construct lattices with quadratic structure over the integers in quadratic number fields having the property that the rank of the quadratic structure is constant and equal to the rank of the lattice in all reductions modulo maximal ideals. We characterize the case in which such lattices are free. The construction gives a representative of the genus of such lattices as an orthogonal sum of "standard" pieces of ranks 1-4 and covers the case of the discriminant of the real quadratic number field congruent to 1 modulo 8 for which a general construction was not known. © 2007 Springer-Verlag.
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| 17. |
- Brzezinski, Juliusz, 1939
(författare)
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What you should know about cubic and quartic equations
- 2011
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Ingår i: Normat - Nordisk Matematisk Tidskrift. - 0801-3500. ; 59:3-4, s. 98-116
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Tidskriftsartikel (refereegranskat)abstract
- The paper describes in a straightforward way how to find Galois groups of cubic and quartic polynomials.
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