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Field theory

Author: Steven Roman
Publisher: New York, N.Y. : Springer, ©2006.
Series: Graduate texts in mathematics, 158.
Edition/Format:   Print book : English : 2nd edView all editions and formats
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"Springer has just released the second edition of Steven Roman's Field Theory, and it continues to be one of the best graduate-level introductions to the subject out there....Every section of the  Read more...

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Genre/Form: Textbooks
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Steven Roman
ISBN: 9780387276779 0387276777 9780387276786 0387276785
OCLC Number: 62395849
Description: xii, 332 pages ; 24 cm.
Contents: Machine derived contents note: 0 Preliminaries 1 --
0.1 L attices --
0.2 Groups 2 --
0.3 The Symmetric Group 10 --
0.4 Rings 10 --
0.5 Integral Domains 14 --
0.6 Unique Factorization Domains 16 --
0.7 Principal Ideal Domains 16 --
0.8 Euclidean Domains 17 --
0.9 Tensor Products 17 --
E xercises 19 --
Part I-Field Extensions --
1 Polynomials 23 --
1.1 Polynomials over a Ring 23 --
1.2 Primitive Polynomials and Irreducibility 24 --
1.3 The Division Algorithm and Its Consequences 27 --
1.4 Splitting Fields 32 --
1.5 The Minimal Polynomial 32 --
1.6 Multiple Roots 33 --
1.7 Testing for Irreducibility 35 --
Exercises 38 --
2 Field Extensions 41 --
2.1 The Lattice of Subfields of a Field 41 --
2.2 Types of Field Extensions 42 --
2.3 Finitely Generated Extensions 46 --
2.4 Simple Extensions 47 --
2.5 Finite Extensions 53 --
2.6 Algebraic Extensions 54 --
2.7 Algebraic Closures 56 --
2.8 Embeddings and Their Extensions 58 --
2.9 Splitting Fields and Normal Extensions 63 --
Exercises 66 --
3 Embeddings and Separability 73 --
3.1 Recap and a Useful Lemma 73 --
3.2 The Number of Extensions: Separable Degree 75 --
3.3 Separable Extensions 77 --
3.4 Perfect Fields 84 --
3.5 Pure Inseparability 85 --
3.6 Separable and Purely Inseparable Closures 88 --
Exercises 91 --
4 Algebraic Independence 93 --
4.1 Dependence Relations 93 --
4.2 Algebraic Dependence 96 --
4.3 Transcendence Bases 100 --
"4.4 Simple Transcendental Extensions 105 --
Exercises 108 --
Part II-Galois Theory --
5 Galois Theory I: An Historical Perspective 113 --
5.1 The Quadratic Equation 113 --
5.2 The Cubic and Quartic Equations 114 --
5.3 Higher-Degree Equations 116 --
5.4 Newton's Contribution: Symmetric Polynomials 117 --
5.5 Vandermonde 119 --
5.6 Lagrange 121 --
5 .7 G au ss 124 --
5.8 B ack to Lagrange 128 --
5 .9 G alois 130 --
5.10 A Very Brief Look at the Life of Galois 135 --
6 Galois Theory II: The Theory 137 --
6.1 G alois C onnections 137 --
6.2 The Galois Correspondence 143 --
6.3 W ho's C losed? 148 --
6.4 Normal Subgroups and Normal Extensions 154 --
6.5 More on Galois Groups 159 --
6.6 Abelian and Cyclic Extensions 164 --
6.7 Linear Disjointness 165 --
Exercises 168 --
7 Galois Theory III: The Galois Group of a Polynomial 173 --
7.1 The Galois Group of a Polynomial 173 --
7.2 Symmetric Polynomials 174 --
7.3 The Fundamental Theorem of Algebra 179 --
7.4 The Discriminant of a Polynomial 180 --
7.5 The Galois Groups of Some Small-Degree Polynomials 182 --
Exercises 193 --
8 A Field Extension as a Vector Space 197 --
8.1 The Norm and the Trace 197 --
8.2 Characterizing Bases 202 --
"8.3 The Normal Basis Theorem 206 --
Exercises 208 --
9 Finite Fields I: Basic Properties 211 --
9.1 Finite Fields Redux 211 --
9.2 Finite Fields as Splitting Fields 212 --
9.3 The Subfields of a Finite Field 213 --
9.4 The Multiplicative Structure of a Finite Field 214 --
9.5 The Galois Group of a Finite Field 215 --
9.6 Irreducible Polynomials over Finite Fields 215 --
"9.7 Normal Bases 218 --
"9.8 The Algebraic Closure of a Finite Field 219 --
Exercises 223 --
10 Finite Fields II: Additional Properties 225 --
10.1 Finite Field Arithmetic 225 --
10.2 The Number of Irreducible Polynomials 232 --
"10.3 Polynomial Functions 234 --
10.4 Linearized Polynomials 236 --
Exercises 238 --
11 The Roots of Unity 239 --
11.1 Roots of Unity 239 --
11.2 Cyclotomic Extensions 241 --
" 11.3 Normal Bases and Roots of Unity 250 --
11.4 Wedderburn's Theorem 251 --
11.5 Realizing Groups as Galois Groups 253 --
E xercises 257 --
12 Cyclic Extensions 261 --
12.1 Cyclic Extensions 261 --
12.2 Extensions of Degree Char(F) 265 --
E xercises 266 --
13 Solvable Extensions 269 --
13.1 Solvable Groups 269 --
13.2 Solvable Extensions 270 --
13.3 Radical Extensions 273 --
13.4 Solvability by Radicals 274 --
13.5 Solvable Equivalent to Solvable by Radicals 276 --
13.6 Natural and Accessory Irrationalities 278 --
13.7 Polynomial Equations 280 --
E xercises 2 82 --
Part Ill-The Theory of Binomials --
14 Binomials 289 --
14.1 Irreducibility 289 --
14.2 The Galois Group of a Binomial 296 --
14.3 The Independence of Irrational Numbers 304 --
E xercises 307 --
15 Families of Binomials 309 --
15.1 The Splitting Field 309 --
15.2 Dual Groups and Pairings 310 --
15.3 Kummer Theory 312 --
Exercises 316 --
Appendix: M dbius Inversion 319 --
Partially Ordered Sets 319 --
The Incidence Algebra of a Partially Ordered Set 320 --
Classical M6bius Inversion 324 --
Multiplicative Version of M6bius Inversion 325.
Series Title: Graduate texts in mathematics, 158.
Responsibility: Steven Roman.
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