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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

I. Galois Theory

Abstract
In this chapter, we develop the machinery of Galois theory. The first four sections constitute the technical heart of Galois theory, and Section 5 presents the fundamental theorem and some consequences. As an application, we give a proof of the fundamental theorem of algebra using Galois theory and the Sylow theorems of group theory.
Patrick Morandi

II. Some Galois Extensions

Abstract
Now that we have developed the machinery of Galois theory, we apply it in this chapter to study special classes of field extensions. Sections 9 and 11 are good examples of how we can use group theoretic information to obtain results in field theory. Section 10 has a somewhat different flavor than the other sections. In it, we look into the classical proof of the Hilbert Theorem 90, a result originally used to help describe cyclic extensions, and from that proof we are led to the study of cohomology, a key tool in algebraic topology, algebraic geometry, and the theory of division rings.
Patrick Morandi

III. Applications of Galois Theory

Abstract
Now that we have developed Galois theory and have investigated a number of types of field extensions, we can put our knowledge to use to answer some of the most famous questions in mathematical history. In Section 15, we look at ruler and compass constructions and prove that with ruler and compass alone it is impossible to trisect an arbitrary angle, to duplicate the cube, to square the circle, and to construct most regular n-gons. These questions arose in the days of the ancient Greeks but were left unanswered for 2500 years. In order to prove that it is impossible to square the circle, we prove in Section 14 that π is transcendental over ℚ, and we prove at the same time that e is also transcendental over ℚ. In Section 16, we prove that there is no algebraic formula, involving only field operations and extraction of roots, to find the roots of an arbitrary nth degree polynomial if n ≥ 5. Before doing so, we investigate in detail polynomials of degree less than 5. By the mid-sixteenth century, formulas for finding the roots of quadratic, cubic, and quartic polynomials had been found. The success in finding the roots of arbitrary cubics and quartics within a few years of each other led people to believe that formulas for arbitrary degree polynomials would be found.
Patrick Morandi

IV. Infinite Algebraic Extensions

Abstract
In this chapter, we investigate infinite Galois extensions and prove an analog of the fundamental theorem of Galois theory for infinite extensions. The key idea is to put a topology on the Galois group of an infinite dimensional Galois extension and then use this topology to determine which subgroups of the Galois group arise as Galois groups of intermediate extensions. We also give a number of constructions of infinite Galois extensions, constructions that arise in quadratic form theory, number theory, and Galois cohomology, among other places.
Patrick Morandi

V. Transcendental Extensions

Abstract
In this chapter, we study field extensions that are not algebraic. In the first two sections, we give the main properties of these extensions. In the remaining sections, we focus on finitely generated extensions. We discuss how these extensions arise in algebraic geometry and how their study can lead to geometric information, and we use algebraic analogs of derivations and differentials to study these extensions.
Patrick Morandi

Backmatter

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