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E-grāmata: Galois Fields and Galois Rings Made Easy

(Claude Bernard University Lyon 1, France)
  • Formāts: PDF+DRM
  • Izdošanas datums: 22-Sep-2017
  • Izdevniecība: ISTE Press Ltd - Elsevier Inc
  • Valoda: eng
  • ISBN-13: 9780081023518
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Galois Fields and Galois Rings Made EasyPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 22-Sep-2017
  • Izdevniecība: ISTE Press Ltd - Elsevier Inc
  • Valoda: eng
  • ISBN-13: 9780081023518
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This book constitutes an elementary introduction to rings and fields, in particular Galois rings and Galois fields, with regard to their application to the theory of quantum information, a field at the crossroads of quantum physics, discrete mathematics and informatics. The existing literature on rings and fields is primarily mathematical. There are a great number of excellent books on the theory of rings and fields written by and for mathematicians, but these can be difficult for physicists and chemists to access. This book offers an introduction to rings and fields with numerous examples. It contains an application to the construction of mutually unbiased bases of pivotal importance in quantum information. It is intended for graduate and undergraduate students and researchers in physics, mathematical physics and quantum chemistry (especially in the domains of advanced quantum mechanics, quantum optics, quantum information theory, classical and quantum computing, and computer engineering). Although the book is not written for mathematicians, given the large number of examples discussed, it may also be of interest to undergraduate students in mathematics.

  • Contains numerous examples that accompany the text
  • Includes an important chapter on mutually unbiased bases
  • Helps physicists and theoretical chemists understand this area of mathematics

Papildus informācija

Presents information on Galois fields and Galois rings in an accessible way so that scientists whose primary area of expertise is not pure mathematics can appreciate
Acknowledgments xi
Preface xiii
List of Mathematical Symbols
xvii
Chapter 1 The Structures of Ring and Field
1(32)
1.1 Rings
1(19)
1.1.1 The ring structure
1(2)
1.1.2 Cardinal of a ring
3(1)
1.1.3 Commutative ring
3(1)
1.1.4 Homomorphism and isomorphism of rings
4(1)
1.1.5 Examples of rings
4(6)
1.1.6 Sub-ring of a ring
10(2)
1.1.7 Ideal of a ring
12(2)
1.1.8 Quotient ring
14(1)
1.1.9 Unitary ring
15(1)
1.1.10 Characteristic of a unitary ring
16(1)
1.1.11 Unit in a unitary ring
17(1)
1.1.12 Zero divisor in a ring
18(2)
1.1.13 Integrity ring
20(1)
1.2 Fields
20(13)
1.2.1 The field structure
20(3)
1.2.2 Cardinal of a field
23(1)
1.2.3 Commutative field
23(1)
1.2.4 Isomorphism and automorphism of fields
23(1)
1.2.5 Examples of fields
24(6)
1.2.6 Sub-field of a field
30(1)
1.2.7 Characteristic of a field
31(2)
Chapter 2 Galois Fields
33(102)
2.1 Generalities
33(4)
2.1.1 Wedderburn's theorem
33(1)
2.1.2 Galois field
34(3)
2.2 Extension of a field: a typical example
37(4)
2.3 Extension of a field: the general case
41(40)
2.3.1 Reducible, irreducible and prime polynomials
42(3)
2.3.2 Examples of (ir)reducible and prime polynomials
45(4)
2.3.3 Quotient field
49(2)
2.3.4 Group structures
51(2)
2.3.5 Primitive element and primitive polynomial
53(5)
2.3.6 Logarithm of a field element
58(1)
2.3.7 Practical rules for constructing a Galois field
59(2)
2.3.8 Examples of extensions of fields
61(15)
2.3.9 Matrix realization of a Galois field
76(5)
2.4 Sub-field of a Galois field
81(1)
2.4.1 GF(pe) sub-field of GF(pm)
81(1)
2.4.2 Characteristic of the sub-fields of GF(pm)
82(1)
2.5 Factorizations
82(13)
2.5.1 Powers of elements of GF(pm)
82(1)
2.5.2 Solutions of εpm - ε = 0
83(3)
2.5.3 Product of all the elements of GF(pm)*
86(1)
2.5.4 Factorization of εpm - ε in prime polynomials
87(3)
2.5.5 Factorization of a prime polynomial
90(5)
2.6 The application trace for a Galois field
95(13)
2.6.1 Trace of an element
95(1)
2.6.2 Frobenius automorphism
96(2)
2.6.3 Elementary properties of the trace
98(6)
2.6.4 Linearity of the trace
104(3)
2.6.5 Trace in terms of the roots of a prime polynomial
107(1)
2.7 Bases of a Galois field
108(10)
2.7.1 Generalities
108(1)
2.7.2 Field bases
109(6)
2.7.3 Dual and self-dual bases
115(3)
2.8 Characters of a Galois field
118(12)
2.8.1 Additive characters
118(7)
2.8.2 Multiplicative characters
125(5)
2.9 Gaussian sums over Galois fields
130(5)
2.9.1 Gauss sum over Zd
130(1)
2.9.2 Quadratic Gauss sum and quadratic characters
131(1)
2.9.3 Gauss sum over GF(pm)
132(1)
2.9.4 Weil sum over GF(pm)
133(2)
Chapter 3 Galois Rings
135(24)
3.1 Generalities
136(1)
3.1.1 Principal ideal of a commutative ring
136(1)
3.1.2 Galois ring
136(1)
3.2 Construction of a Galois ring
137(8)
3.2.1 Elements of Zps
137(1)
3.2.2 The Zps → Zp and Zps [ ε] → Zp[ ε] homomorphisms
138(2)
3.2.3 Basic irreducible polynomial
140(1)
3.2.4 Extension of a base ring
141(1)
3.2.5 Isomorphism of two Galois rings
142(1)
3.2.6 Sub-ring of a Galois ring
143(1)
3.2.7 Adic (p-adic) decomposition
143(2)
3.3 Examples and counter-examples of Galois rings
145(8)
3.3.1 Counter-examples
145(4)
3.3.2 Examples
149(4)
3.4 The application trace for a Galois ring
153(2)
3.4.1 Generalized Frobenius automorphism and trace
153(1)
3.4.2 Elementary properties of the trace
154(1)
3.5 Characters of a Galois ring
155(1)
3.6 Gaussian sums over Galois rings
156(3)
3.6.1 Gauss sum over GR(ps, m)
156(1)
3.6.2 Weil sum over GR(ps, m)
156(3)
Chapter 4 Mutually Unbiased Bases
159(40)
4.1 Generalities
161(4)
4.1.1 Unbiased bases
161(1)
4.1.2 Example: d = 2
161(1)
4.1.3 Interests of MUBs for quantum mechanics
162(1)
4.1.4 Well-known results
163(2)
4.2 Quantum angular momentum bases
165(7)
4.2.1 Standard basis for SU(2)
165(2)
4.2.2 Non-standard bases for SU(2)
167(1)
4.2.3 Bases in quantum information
168(4)
4.3 SU(2) approach to mutually unbiased bases
172(17)
4.3.1 A master formula for d = p (p prime)
172(3)
4.3.2 Examples: d = 2 and 3
175(2)
4.3.3 An alternative formula for d = p (p odd prime)
177(1)
4.3.4 Weyl pairs
177(6)
4.3.5 MUBs and the special linear group
183(1)
4.3.6 MUBs for d power of a prime
184(5)
4.4 Galois field approach to mutually unbiased bases
189(5)
4.4.1 Weyl pair for GF(pm)
190(1)
4.4.2 Bases in the frame of GF(pm)
191(2)
4.4.3 MUBs in the frame of GF(pm)
193(1)
4.5 Galois ring approach to mutually unbiased bases
194(5)
4.5.1 Bases in the frame of GR(22, m)
194(1)
4.5.2 MUBs in the frame of GR(22, m)
195(1)
4.5.3 One-and two-qubit systems
196(3)
Chapter 5 Appendix on Number Theory and Group Theory
199(34)
5.1 Elements of number theory
199(15)
5.1.1 Euler function
199(1)
5.1.2 Mobius function
200(1)
5.1.3 Root of unity
201(2)
5.1.4 Cyclotomic polynomials
203(2)
5.1.5 Residue
205(1)
5.1.6 Quadratic residue
206(4)
5.1.7 Gauss sums
210(4)
5.2 Elements of group theory
214(19)
5.2.1 Axioms of group
214(1)
5.2.2 Direct product of groups
215(1)
5.2.3 Homomorphism, isomorphism and automorphism of groups
216(1)
5.2.4 Conjugate classes
216(1)
5.2.5 Sub-group
217(1)
5.2.6 Cyclic group
217(1)
5.2.7 Cosets
218(1)
5.2.8 Lagrange's theorem
219(1)
5.2.9 Order of a group element
219(1)
5.2.10 Quotient group
220(1)
5.2.11 Abstract group - group table
220(2)
5.2.12 Examples of groups
222(3)
5.2.13 Representations of a group
225(2)
5.2.14 Orthogonality relations
227(6)
Bibliography 233(10)
Index 243
Maurice Kibler is Professor Emeritus at Claude Bernard University Lyon 1 in France. His research concerns the role of symmetries in the elaboration of models in various domains of physics (sub-atomic, atomic, molecular and condensed matter physics). He is also interested in quantum information.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
Permanent link: https://www.kriso.lv/db/97800810235182e.html
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