Abel's irreducibility theorem (original) (raw)
In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F. Corollaries of the theorem include:
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| dbo:abstract | In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F. Corollaries of the theorem include: * If ƒ(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 − 2 is irreducible over the rational numbers and has as a root; hence there is no linear or constant polynomial over the rationals having as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x). * If ƒ(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots. (en) |
| dbo:wikiPageExternalLink | http://fermatslasttheorem.blogspot.com/2008/09/abels-lemmas-on-irreducibility.html |
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| dbo:wikiPageWikiLink | dbr:Monic_polynomial dbr:Degree_of_a_polynomial dbr:Mathematics dbr:Corollaries dbr:Irreducible_polynomial dbr:Field_(mathematics) dbr:Niels_Henrik_Abel dbc:Field_(mathematics) dbr:Coefficient dbr:Polynomial dbr:Field_theory_(mathematics) dbr:Rational_number dbr:Root_of_a_polynomial dbr:Zero_polynomial |
| dbp:title | Abel's Irreducibility Theorem (en) |
| dbp:urlname | AbelsIrreducibilityTheorem (en) |
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| dct:subject | dbc:Field_(mathematics) |
| rdfs:comment | In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F. Corollaries of the theorem include: (en) |
| rdfs:label | Abel's irreducibility theorem (en) |
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