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[Texts]## 12F: Field extensions |

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.

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.

- 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.

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

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]

- What are the conditions on the coefficients of a polynomial for all roots to be rational?
- Proof of the Rational Roots Theorem
- Will some variant of Eisenstein's Criterion always suffice to prove irreducibility? (no)
- Citation: is an integer polynomial a product of cyclotomic polynomials?
- Does a particular polynomial have roots of unity among its roots?
- Example of the Galois correspondence for infinite groups
- An example from Galois theory: calculating the fixed field K(X)^G, for a certain small G.
- Example of the fixed field under a subgroup of the Galois group ( = Sym(3) ).
- A computer algebra challenge: to find subfields of a certain extension of Q.
- Some nested radical expressions simplify
- Solution of cubic equations (and higher)
- If a root of a cubic is rational, must sqrt(disc) be rational, too? (no)
- Deriving solutions to quartic equations in radicals
- Which quintics
*can*be solved by radicals? - Abel's original paper on the general quintic
- Closed-form (non-radical) solutions of quintic
- Pointer to Dummit's article on solving solvable quintics, and other background information about the impossibility of solving a general quintic polynomial with radicals. [For solutions using other functions see 26C.]
- Interesting example of Galois groups of certain sextics.
- Computing a Galois group (of a sextic whose group is not the symmetric group).
- Sample use of Galois theory to solve a particular octic polynomial in radicals
- How may Galois groups be computed?
- If the Galois group is solvable, one can express the roots with the standard operations. (Citations)
- Almost all Galois groups are the symmetric group.
- Is the Galois group of random polynomial the symmetric group?
- A citation to the irreducibility of trinomials of the form x^a + x^b + 1
- Irreducibility of trinomials of the form x^a + A x^b + B
- How to make irreducible polynomials over a finite field, of large degree
- Constructing irreducible polynomials over finite fields -- Citation
- Testing irreducibility of polynomials over finite fields - Berlekamp's algorithm (with citations).
- Testing polynomials (mod p) for irreducibility.
- Heuristic irreducibility test for polynomials in Z[x]
- How can one decide if a polynomial is irreducible (here, over F_p).
- How does one factor polynomials in 1 variable?
- How does one factor polynomials in 1 variable? -- take 2
- Factoring polynomials over finite fields
- Relative merits of methods of factorization of polynomials in (Z/pZ)[x]
- Factoring in polynomial rings over finite fields
- Current trends in factorization of integer polynomials
- Citation: how do computers factor in Z[X]?
- Factoring the polynomial x^500+x+1 modulo a 152-digit prime. (with citations)
- Can you factor a linear combination of polynomials of the form a^2+b^2+1 (a, b linear in x,y)?
- Using HGCD to compute common divisors of rational polynomials
- Alternative ways to divide polynomials (divide P1/P2 if you know the roots of P2).
- Are roots of polynomials over Q dense in C^n? Yes, by the Hilbert Irreducibility Theorem
- The irreducible rational polynomials are dense in Q[X].
- How do Galois groups vary with the coefficients of the polynomial?
- Determining irreducibility of a multivariate polynomial by many specializations of one variable: the Hilbert Irreducibility Theorem.
- How to compute (a primitive element for) the splitting field of a polynomial? (Maple example.)
- How to tell if a family of polynomials is linearly independent over Q.
- Citation for chestnut: cubics with real roots are not in a real-radical extension.
- An irreducible polynomial solvable in real radicals must be of degree a power of two. (Pointer)
- Illustration of Galois theory as it pertains to certain values of trig functions.
- When can cos(p*Pi/q) be expressed with real radicals?
- Solving equations in radicals when the Galois group is cyclic.
- General approach to algebraic solutions of polynomial equations: use of the Lagrange resolvent.
- Here's a theorem half-way to algebraic geometry or elliptic curves: if P is a quintic, there are 80 cubics y such that y^2-P is a perfect cube (Noam Elkies)
- Predicting the Galois group of a polynomial using Cebotarev's Theorem (f splits mod p for 1/|G| of all primes p)
- Galois group actions without n-cycles in action on roots of a polynomial of degree n
- The inverse Galois problem: showing that every finite group is the Galois group of some number field (with example for G = Z/3Z )
- The Inverse Galois Theory problem -- is every group a Galois group?

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