Lectures on finite fields and galois rings:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2003
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturver. S. [335] - 337 |
| Physical Description: | X, 342 S. |
| ISBN: | 9812385045 9812385703 |
Staff View
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Record in the Search Index
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| adam_text | O D E [;E ] ;I ; K MN CYP O , LECTURES ON FINITE FIELDS AND GALOIS
RINGS ZHE-XIAN WAN C/I/NESE ACADEMY OF SCIENCES, CHINA WORLD SCIENTIFIC
NEW JERSEY * LONDON * SINGAPORE * HONG KONG CONTENTS 1 SETS AND INTEGERS
1 1.1 SETS AND MAPS 1 1.2 THE FACTORIZATION OF INTEGERS 7 1.3
EQUIVALENCE RELATION AND PARTITION 15 1.4 EXERCISES 18 2 GROUPS 21 2.1
THE CONCEPT OF A GROUP AND EXAMPLES 21 2.2 SUBGROUPS AND COSETS 31 2.3
CYCLIC GROUPS 38 2.4 EXERCISES 45 3 FIELDS AND RINGS 49 3.1 FIELDS 49
3.2 THE CHARACTERISTIC OF A FIELD 58 3.3 RINGS AND INTEGRAL DOMAINS 64
3.4 FIELD OF FRACTIONS OF AN INTEGRAL DOMAIN 67 3.5 DIVISIBILITY IN A
RING 70 3.6 EXERCISES 72 4 POLYNOMIALS 75 4.1 POLYNOMIAL RINGS 75 4.2
DIVISION ALGORITHM 80 * VII VIII CONTENTS 4.3 EUCLIDEAN ALGORITHM 83 4.4
UNIQUE FACTORIZATION OF POLYNOMIALS 93 4.5 EXERCISES 99 5 RESIDUE CLASS
RINGS 101 5.1 RESIDUE CLASS RINGS 101 5.2 EXAMPLES 106 5.3 RESIDUE CLASS
FIELDS 108 5.4 MORE EXAMPLES ILL 5.5 EXERCISES 114 6 STRUCTURE OF FINITE
FIELDS 115 6.1 THE MULTIPLICATIVE GROUP OF A FINITE FIELD 115 6.2 THE
NUMBER OF ELEMENTS IN A FINITE FIELD 120 6.3 EXISTENCE OF FINITE FIELD
WITH P N ELEMENTS 122 6.4 UNIQUENESS OF FINITE FIELD WITH P N ELEMENTS
127 6.5 SUBFIELDS OF FINITE FIELDS 128 6.6 A DISTINCTION BETWEEN FINITE
FIELDS OF CHARACTERISTIC 2 AND NOT 2 130 6.7 EXERCISES 133 7 FURTHER
PROPERTIES OF FINITE FIELDS 137 7.1 AUTOMORPHISMS 137 7.2 CHARACTERISTIC
POLYNOMIALS AND MINIMAL POLYNOMIALS 140 7.3 PRIMITIVE POLYNOMIALS 145
7.4 TRACE AND NORM 148 7.5 QUADRATIC EQUATIONS 155 7.6 EXERCISES 158 8
BASES 161 8.1 BASES AND POLYNOMIAL BASES 161 8.2 DUAL BASES 166 CONTENTS
IX 8.3 SELF-DUAL BASIS 173 8.4 NORMAL BASES 180 8.5 EXERCISES 189 9
FACTORING POLYNOMIALS OVER FINITE FIELDS 191 9.1 FACTORING POLYNOMIALS
OVER FINITE FIELDS 191 9.2 FACTORIZATION OF X N - 1 202 9.3 CYCLOTOMIC
POLYNOMIALS 206 9.4 THE PERIOD OF A POLYNOMIAL 210 9.5 EXERCISES 216 10
IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS 219 10.1 ON THE DETERMINATION
OF IRREDUCIBLE POLYNOMIALS 219 10.2 IRREDUCIBILITY CRITERION OF
BINOMIALS 221 10.3 SOME IRREDUCIBLE TRINOMIALS 225 10.4 COMPOSITIONS OF
POLYNOMIALS 231 10.5 RECURSIVE CONSTRUCTIONS 237 10.6 COMPOSED PRODUCT
AND SUM OF POLYNOMIALS 241 10.7 IRREDUCIBLE POLYNOMIALS OF ANY DEGREE
244 10.8 EXERCISES 246 11 QUADRATIC FORMS OVER FINITE FIELDS 249 11.1
QUADRATIC FORMS OVER FINITE FIELDS OF CHARACTERISTIC NOT 2 . 249 11.2
ALTERNATE FORMS OVER FINITE FIELDS 255 11.3 QUADRATIC FORMS OVER FINITE
FIELDS OF CHARACTERISTIC 2 .... 258 11.4 EXERCISES 268 12 MORE GROUP
THEORY AND RING THEORY 271 12.1 HOMOMORPHISMS OF GROUPS, NORMAL
SUBGROUPS AND FACTOR GROUPS 271 12.2 DIRECT PRODUCT DECOMPOSITION OF
GROUPS 279 12.3 SOME RING THEORY 284 X CONTENTS 12.4 MODULES 292 12.5
EXERCISES 295 13 HENSEL S LEMMA AND HENSEL LIFT 297 13.1 THE POLYNOMIAL
RING Z P .[X] 297 13.2 HENSEL S LEMMA 300 13.3 FACTORIZATION OF MONIC
POLYNOMIALS IN 7L V X 30 2 13.4 BASIC IRREDUCIBLE POLYNOMIALS AND
HENSEL LIFT 304 13.5 EXERCISES 308 14 GALOIS RINGS 309 14.1 EXAMPLES OF
GALOIS RINGS 309 14.2 STRUCTURE OF GALOIS RINGS 313 14.3 THE P-ADIC
REPRESENTATION 316 14.4 THE GROUP OF UNITS OF A GALOIS RING 319 14.5
EXTENSION OF GALOIS RINGS 323 14.6 AUTOMORPHISMS OF GALOIS RINGS 327
14.7 GENERALIZED TRACE AND NORM 331 14.8 EXERCISES 332 BIBLIOGRAPHY 335
INDEX 339
|
| adam_txt |
O D E [;E ] ;I ; 'K ' MN'CYP O , LECTURES ON FINITE FIELDS AND GALOIS
RINGS ZHE-XIAN WAN C/I/NESE ACADEMY OF SCIENCES, CHINA WORLD SCIENTIFIC
NEW JERSEY * LONDON * SINGAPORE * HONG KONG CONTENTS 1 SETS AND INTEGERS
1 1.1 SETS AND MAPS 1 1.2 THE FACTORIZATION OF INTEGERS 7 1.3
EQUIVALENCE RELATION AND PARTITION 15 1.4 EXERCISES 18 2 GROUPS 21 2.1
THE CONCEPT OF A GROUP AND EXAMPLES 21 2.2 SUBGROUPS AND COSETS 31 2.3
CYCLIC GROUPS 38 2.4 EXERCISES 45 3 FIELDS AND RINGS 49 3.1 FIELDS 49
3.2 THE CHARACTERISTIC OF A FIELD 58 3.3 RINGS AND INTEGRAL DOMAINS 64
3.4 FIELD OF FRACTIONS OF AN INTEGRAL DOMAIN 67 3.5 DIVISIBILITY IN A
RING 70 3.6 EXERCISES 72 4 POLYNOMIALS 75 4.1 POLYNOMIAL RINGS 75 4.2
DIVISION ALGORITHM 80 * VII VIII CONTENTS 4.3 EUCLIDEAN ALGORITHM 83 4.4
UNIQUE FACTORIZATION OF POLYNOMIALS 93 4.5 EXERCISES 99 5 RESIDUE CLASS
RINGS 101 5.1 RESIDUE CLASS RINGS 101 5.2 EXAMPLES 106 5.3 RESIDUE CLASS
FIELDS 108 5.4 MORE EXAMPLES ILL 5.5 EXERCISES 114 6 STRUCTURE OF FINITE
FIELDS 115 6.1 THE MULTIPLICATIVE GROUP OF A FINITE FIELD 115 6.2 THE
NUMBER OF ELEMENTS IN A FINITE FIELD 120 6.3 EXISTENCE OF FINITE FIELD
WITH P N ELEMENTS 122 6.4 UNIQUENESS OF FINITE FIELD WITH P N ELEMENTS
127 6.5 SUBFIELDS OF FINITE FIELDS 128 6.6 A DISTINCTION BETWEEN FINITE
FIELDS OF CHARACTERISTIC 2 AND NOT 2 130 6.7 EXERCISES 133 7 FURTHER
PROPERTIES OF FINITE FIELDS 137 7.1 AUTOMORPHISMS 137 7.2 CHARACTERISTIC
POLYNOMIALS AND MINIMAL POLYNOMIALS 140 7.3 PRIMITIVE POLYNOMIALS 145
7.4 TRACE AND NORM 148 7.5 QUADRATIC EQUATIONS 155 7.6 EXERCISES 158 8
BASES 161 8.1 BASES AND POLYNOMIAL BASES 161 8.2 DUAL BASES 166 CONTENTS
IX 8.3 SELF-DUAL BASIS 173 8.4 NORMAL BASES 180 8.5 EXERCISES 189 9
FACTORING POLYNOMIALS OVER FINITE FIELDS 191 9.1 FACTORING POLYNOMIALS
OVER FINITE FIELDS 191 9.2 FACTORIZATION OF X N - 1 202 9.3 CYCLOTOMIC
POLYNOMIALS 206 9.4 THE PERIOD OF A POLYNOMIAL 210 9.5 EXERCISES 216 10
IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS 219 10.1 ON THE DETERMINATION
OF IRREDUCIBLE POLYNOMIALS 219 10.2 IRREDUCIBILITY CRITERION OF
BINOMIALS 221 10.3 SOME IRREDUCIBLE TRINOMIALS 225 10.4 COMPOSITIONS OF
POLYNOMIALS 231 10.5 RECURSIVE CONSTRUCTIONS 237 10.6 COMPOSED PRODUCT
AND SUM OF POLYNOMIALS 241 10.7 IRREDUCIBLE POLYNOMIALS OF ANY DEGREE
244 10.8 EXERCISES 246 11 QUADRATIC FORMS OVER FINITE FIELDS 249 11.1
QUADRATIC FORMS OVER FINITE FIELDS OF CHARACTERISTIC NOT 2 . 249 11.2
ALTERNATE FORMS OVER FINITE FIELDS 255 11.3 QUADRATIC FORMS OVER FINITE
FIELDS OF CHARACTERISTIC 2 . 258 11.4 EXERCISES 268 12 MORE GROUP
THEORY AND RING THEORY 271 12.1 HOMOMORPHISMS OF GROUPS, NORMAL
SUBGROUPS AND FACTOR GROUPS 271 12.2 DIRECT PRODUCT DECOMPOSITION OF
GROUPS 279 12.3 SOME RING THEORY 284 X CONTENTS 12.4 MODULES 292 12.5
EXERCISES 295 13 HENSEL'S LEMMA AND HENSEL LIFT 297 13.1 THE POLYNOMIAL
RING Z P .[X] 297 13.2 HENSEL'S LEMMA 300 13.3 FACTORIZATION OF MONIC
POLYNOMIALS IN 7L V \X\ 30 2 13.4 BASIC IRREDUCIBLE POLYNOMIALS AND
HENSEL LIFT 304 13.5 EXERCISES 308 14 GALOIS RINGS 309 14.1 EXAMPLES OF
GALOIS RINGS 309 14.2 STRUCTURE OF GALOIS RINGS 313 14.3 THE P-ADIC
REPRESENTATION 316 14.4 THE GROUP OF UNITS OF A GALOIS RING 319 14.5
EXTENSION OF GALOIS RINGS 323 14.6 AUTOMORPHISMS OF GALOIS RINGS 327
14.7 GENERALIZED TRACE AND NORM 331 14.8 EXERCISES 332 BIBLIOGRAPHY 335
INDEX 339 |
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| dewey-ones | 512 - Algebra |
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| discipline | Mathematik |
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| id | DE-604.BV022199601 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:23:58Z |
| indexdate | 2024-07-09T20:52:13Z |
| institution | BVB |
| isbn | 9812385045 9812385703 |
| language | English |
| lccn | 2003278789 |
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| physical | X, 342 S. |
| publishDate | 2003 |
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| publisher | World Scientific |
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| spelling | Wan, Zhexian 1927- Verfasser (DE-588)132646943 aut Lectures on finite fields and galois rings Zhe-Xian Wan Singapore [u.a.] World Scientific 2003 X, 342 S. txt rdacontent n rdamedia nc rdacarrier Literaturver. S. [335] - 337 aFinite fields (Algebra) aGalois theory Galois-Feld (DE-588)4155896-0 gnd rswk-swf Galois-Feld (DE-588)4155896-0 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015411056&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wan, Zhexian 1927- Lectures on finite fields and galois rings aFinite fields (Algebra) aGalois theory Galois-Feld (DE-588)4155896-0 gnd |
| subject_GND | (DE-588)4155896-0 |
| title | Lectures on finite fields and galois rings |
| title_auth | Lectures on finite fields and galois rings |
| title_exact_search | Lectures on finite fields and galois rings |
| title_exact_search_txtP | Lectures on finite fields and galois rings |
| title_full | Lectures on finite fields and galois rings Zhe-Xian Wan |
| title_fullStr | Lectures on finite fields and galois rings Zhe-Xian Wan |
| title_full_unstemmed | Lectures on finite fields and galois rings Zhe-Xian Wan |
| title_short | Lectures on finite fields and galois rings |
| title_sort | lectures on finite fields and galois rings |
| topic | aFinite fields (Algebra) aGalois theory Galois-Feld (DE-588)4155896-0 gnd |
| topic_facet | aFinite fields (Algebra) aGalois theory Galois-Feld |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015411056&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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