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In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a, b, and c are indeterminates, the generic polynomial of degree two in 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, ..., tn) 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 so

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rdf:type	album
rdfs:label	Generic polynomial (en)
rdfs:comment	In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a, b, and c are indeterminates, the generic polynomial of degree two in 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, ..., tn) 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 so (en)
dcterms:subject	Field (mathematics) Galois theory
Wikipage page ID	1400173 (xsd:integer)
Wikipage revision ID	1105723004 (xsd:integer)
Link from a Wikipage to another Wikipage	Monic polynomial Indeterminate (variable) Inverse Galois problem Mathematics Galois theory Parity (mathematics) Galois group Heisenberg group Algebra Alternating group Cyclic group Field (mathematics) Quaternion group Wreath product Hendrik Lenstra Prime number Field (mathematics) Galois theory Coefficient DIH Direct product of groups Polynomial Splitting field Symmetric group Finite group Field of rational functions Divisibility
sameAs	Generic polynomial Generic polynomial Generic polynomial
dbp:wikiPageUsesTemplate	dbt:Math
has abstract	In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a, b, and c are indeterminates, the generic polynomial of degree two in 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, ..., tn) 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. (en)
gold:hypernym	P
prov:wasDerivedFrom	wikipedia-en:Generic_polynomial?oldid=1105723004&ns=0
page length (characters) of wiki page	4326 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Generic_polynomial
is Link from a Wikipage to another Wikipage of	Resultant Inverse Galois problem Bézout's theorem Generic Discriminant
is Wikipage disambiguates of	Generic
is foaf:primaryTopic of	wikipedia-en:Generic_polynomial

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