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| In [[abstract algebra]], a '''rupture field''' of a [[polynomial]] <math>P(X)</math> over a given [[field (mathematics)|field]] <math>K</math> such that <math>P(X)\in K[X]</math> is the [[field extension]] of <math>K</math> generated by a [[root of a function|root]] <math>a</math> of <math>P(X)</math>.<ref>{{Cite book
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| | last = Escofier
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| | first = Jean-Paul
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| | title = Galois Theory
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| | publisher = Springer
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| | date = 2001
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| | pages = 62
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| | isbn = 0-387-98765-7}}
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| </ref>
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| For instance, if <math>K=\mathbb Q</math> and <math>P(X)=X^3-2</math> then <math>\mathbb Q[\sqrt[3]2]</math> is a rupture field for <math>P(X)</math>.
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| The notion is interesting mainly if <math>P(X)</math> is [[irreducible polynomial|irreducible]] over <math>K</math>. In that case, all rupture fields of <math>P(X)</math> over <math>K</math> are isomorphic, non canonically, to <math>K_P=K[X]/(P(X))</math>: if <math>L=K[a]</math> where <math>a</math> is a root of <math>P(X)</math>, then the [[ring homomorphism]] <math>f</math> defined by <math>f(k)=k</math> for all <math>k\in K</math> and <math>f(X\mod P)=a</math> is an [[isomorphism]]. Also, in this case the degree of the extension equals the degree of <math>P</math>.
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| The rupture field of a [[polynomial]] does not necessarily contain all the roots of that [[polynomial]]: in the above example the field <math>\mathbb Q[\sqrt[3]2]</math> does not contain the other two (complex) roots of <math>P(X)</math> (namely <math>\omega\sqrt[3]2</math> and <math>\omega^2\sqrt[3]2</math> where <math>\omega</math> is a primitive third root of unity). For a field containing all the roots of a [[polynomial]], see the [[splitting field]].
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| ==Examples==
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| The rupture field of <math>X^2+1</math> over <math>\mathbb R</math> is <math>\mathbb C</math>. It is also its [[splitting field]].
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| The rupture field of <math>X^2+1</math> over <math>\mathbb F_3</math> is <math>\mathbb F_9</math> since there is no element of <math>\mathbb F_3</math> with square equal to <math>-1</math> (and all quadratic [[field extension|extensions]] of <math>\mathbb F_3</math> are isomorphic to <math>\mathbb F_9</math>).
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| ==See also==
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| * [[Splitting field]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Rupture Field}}
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| [[Category:Field theory]]
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Latest revision as of 18:06, 5 January 2015
Bricklayer Darell from Rockland, likes to spend some time acting, como ganhar dinheiro na internet and car. Enjoys travel and ended up encouraged after visiting Notre-Dame Cathedral in Tournai.
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