In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a generic polynomial refers usually to a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
whose
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s are
indeterminates. For example, if , , and are indeterminates, the generic polynomial of degree two in is
However in
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, a branch of
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, and in this article, the term ''generic polynomial'' has a different, although related, meaning: a generic polynomial for a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
''G'' and a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''F'' is a
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\cd ...
''P'' with coefficients in the
field of rational functions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
''L'' = ''F''(''t''
1, ..., ''t''
''n'') in ''n'' indeterminates over ''F'', such that the
splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
''M'' of ''P'' has
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
''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
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved.
There ...
for that group. However, not all Galois groups have generic polynomials, a counterexample being the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order eight.
Groups with generic polynomials
* The
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
''S''
''n''. This is trivial, as
:
:is a generic polynomial for ''S''
''n''.
* Cyclic groups ''C''
''n'', where ''n'' is not
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
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
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
''D''
''n'' has a generic polynomial if and only if ''n'' is not divisible by eight.
* The
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
''Q''
8.
*
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ' ...
s
for any
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p''.
* The
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
''A''
4.
* The alternating group ''A''
5.
* Reflection groups defined over Q, including in particular groups of the root systems for ''E''
6, ''E''
7, and ''E''
8.
* Any group which is a
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of two groups both of which have generic polynomials.
* Any group which is a
wreath product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used i ...
of two groups both of which have generic polynomials.
Examples of generic polynomials
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
Field (mathematics)
Galois theory