“Generic polynomial”的意思、由来-开放百科全书

词条

Generic polynomial

释义

Groups with generic polynomials
Examples of generic polynomials
Generic Dimension
Publications

In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if {{math|a}}, {{math|b}}, and {{math|c}} are indeterminates, the generic polynomial of degree two in {{math|x}} is

However in Galois theory, a branch of algebra, and in this article, the term generic polynomial has a different, although related, meaning: a generic polynomial for a finite group G and a field F is a monic polynomial P with coefficients in the field of rational functions L = F(t₁, ..., t_n) in n indeterminates over F, such that the splitting field M of P has Galois group G over L, and such that every extension K/F with Galois group G can be obtained as the splitting field of a polynomial which is the specialization of P resulting from setting the n indeterminates to n elements of F. This is sometimes called F-generic or relative to the field F; a Q-generic polynomial, which is generic relative to the rational numbers is called simply generic.

The existence, and especially the construction, of a generic polynomial for a given Galois group provides a complete solution to the inverse Galois problem for that group. However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of order eight.

Groups with generic polynomials

The symmetric group S_n. This is trivial, as

is a generic polynomial for S_n.

Cyclic groups C_n, where n is not divisible by eight. Lenstra showed that a cyclic group does not have a generic polynomial if n is divisible by eight, and G. W. Smith explicitly constructs such a polynomial in case n is not divisible by eight.
The cyclic group construction leads to other classes of generic polynomials; in particular the dihedral group D_n has a generic polynomial if and only if n is not divisible by eight.
The quaternion group Q₈.
Heisenberg groups for any odd prime p.
The alternating group A₄.
The alternating group A₅.
Reflection groups defined over Q, including in particular groups of the root systems for E₆, E₇, and E₈.
Any group which is a direct product of two groups both of which have generic polynomials.
Any group which is a wreath product of two groups both of which have generic polynomials.

Examples of generic polynomials

Group	Generic Polynomial
C₂
C₃
S₃
V
C₄
D₄
S₄
D₅
S₅

Generic polynomials are known for all transitive groups of degree 5 or less.

Generic Dimension

The generic dimension for a finite group G over a field F, denoted , is defined as the minimal number of parameters in a generic polynomial for G over F, or if no generic polynomial exists.

Examples:

Publications

Jensen, Christian U., Ledet, Arne, and Yui, Noriko, Generic Polynomials, Cambridge University Press, 2002

2 : Field theory|Galois theory

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。