Field and Galois Theory

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Springer Science & Business Media, Jul 25, 1996 - Mathematics - 281 pages
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In the fall of 1990, I taught Math 581 at New Mexico State University for the first time. This course on field theory is the first semester of the year-long graduate algebra course here at NMSU. In the back of my mind, I thought it would be nice someday to write a book on field theory, one of my favorite mathematical subjects, and I wrote a crude form of lecture notes that semester. Those notes sat undisturbed for three years until late in 1993 when I finally made the decision to turn the notes into a book. The notes were greatly expanded and rewritten, and they were in a form sufficient to be used as the text for Math 581 when I taught it again in the fall of 1994. Part of my desire to write a textbook was due to the nonstandard format of our graduate algebra sequence. The first semester of our sequence is field theory. Our graduate students generally pick up group and ring theory in a senior-level course prior to taking field theory. Since we start with field theory, we would have to jump into the middle of most graduate algebra textbooks. This can make reading the text difficult by not knowing what the author did before the field theory chapters. Therefore, a book devoted to field theory is desirable for us as a text. While there are a number of field theory books around, most of these were less complete than I wanted.
  

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Contents

Galois Theory
1
2 Automorphisms
15
3 Normal Extensions
27
4 Separable and Inseparable Extensions
39
5 The Fundamental Theorem of Galois Theory
51
Some Galois Extensions
65
7 Cyclotomic Extensions
71
8 Norms and Traces
78
23 Derivations and Differentials
210
Ring Theory
225
1 Prime and Maximal Ideals
226
2 Unique Factorization Domains
227
3 Polynomials over a Field
230
4 Factorization in Polynomial Rings
232
5 Irreducibility Tests
234
Set Theory
241

9 Cyclic Extensions
87
10 Hilbert Theorem 90 and Group Cohomology
93
11 Kummer Extensions
104
Applications of Galois Theory
111
12 Discriminants
112
13 Polynomials of Degree 3 and 4
123
14 The Transcendence of 𝜋 and e
133
15 Ruler and Compass Constructions
140
16 Solvability by Radicals
147
Infinite Algebraic Extensions
155
18 Some Infinite Galois Extensions
164
Transcendental Extensions
173
20 Linear Disjointness
182
21 Algebraic Varieties
192
22 Algebraic Function Fields
201
2 Cardinality and Cardinal Arithmetic
243
Group Theory
245
2 The Sylow Theorems
247
3 Solvable Groups
248
4 Profinite Groups
249
Vector Spaces
255
2 Linear Transformations
257
3 Systems of Linear Equations and Determinants
260
4 Tensor Products
261
Topology
267
2 Topological Properties
270
References
275
Index
277
Copyright

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