## Field and Galois TheoryIn 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 |

275 | |

277 | |

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### Common terms and phrases

algebraic closure algebraic extension algebraic over F algebraically independent automorphism basis for K/F char(F closure of F cocycle constructible Corollary defined definition discriminant disjoint over F divides element Example extension K/F extension of F F-homomorphism F-vector fc-variety field extension field F field of characteristic field of f finite extension fixed field fundamental theorem Gal(K/F Galois extension Galois group Galois over F Galois theory hence homomorphism independent over F integer intermediate field irreducible over F irreducible polynomial Let F Let f(x Let G linearly disjoint matrix maximal min(F minimal polynomial nonzero nth root prime ideal primitive nth root Problem profinite group Proof Proposition prove purely inseparable radical extension ring root of f root of unity Section separable extension separable over F solvable splitting field subfield subgroup of G subset Suppose topology transcendence basis transcendental vector space Zorn's lemma