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# 12F: Field extensions

## Introduction

This is the appropriate page for Galois Theory.

Once upon a time, mathematicians (and others) would spend time on a subject call the "Theory of equations", which was just chock-full of algorithms and the theory of polynomials and their roots. Highly recommend is an old book by Uspensky as typical of this genre. Nowadays, this is the subject of ring theory or numerical analysis, but we chose to keep much of that material here since it often involves a consideration of the splitting fields of that polynomial.

## Applications and related fields

This page is appropriate for the algebraic study of a single polynomial. The general algebraic inspection of an integral polynomial could appropriately occur in many subfields of number theory (11) or field theory (12). For example, one may look at the extension field it generates in algebraic number theory.

The algebraic study of general collections of polynomials (e.g. Gröbner bases) is appropriate for commutative ring theory. Specific families of polynomials, e.g. the Chebyshev polynomials, are treated in Special functions, or Number Theory or Fourier series (and orthogonal functions) as appropriate.

One may seek the roots of a polynomial numerically in Numerical Analysis

We view polynomials as special types of (analytic) functions in Real analysis or Complex analysis. In particular, in the page for Real polynomials we find Descartes' rule of signs, Sturm sequences, the solutions to cubic and quartic polynomials, Hilbert's 17th problem, and so on, since those topics are not principally concerned with the algebraic aspect of the solutions.

Polynomials in several variables are used to define varieties in algebraic geometry. Indeed, algebraic varieties give a geometric perspective on the sets of solutions of polynomial equations.

## Subfields

• 12F05: Algebraic extensions
• 12F10: Separable extensions, Galois theory
• 12F12: Inverse Galois theory
• 12F15: Inseparable extensions
• 12F20: Transcendental extensions
• 12F99: None of the above but in this section

Parent field: 12: Field theory and polynomials

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

For a pleasant introduction to Galois theory see Hadlock, Charles Robert: "Field theory and its classical problems", Carus Mathematical Monographs, 19. Mathematical Association of America, Washington, D.C., 1978. 323 pp. ISBN 0-88385-020-6 MR82c:12001

Current status of the Inverse Galois Problem: Matzat, B. Heinrich: "Der Kenntnisstand in der konstruktiven Galoisschen Theorie" (German; "State of the art in constructive Galois theory") Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 65--98, Progr. Math., 95, Birkhäuser, Basel, 1991. MR93g:12005

## Software and tables

Announcement: Table of number fields data [Henri Cohen]

Software pointer: NTL: a C++ library for bignums and algebra over Z and finite fields [Victor Shoup]

## Selected topics at this site

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Last modified 2000/01/14 by Dave Rusin. Mail: