Splitting field: Difference between revisions
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In [[algebra]], a '''splitting field''' for a polynomial ''f'' over a field ''F'' is a [[field extension]] ''E''/''F'' with the properties that ''f'' splits completely over ''E'', but | In [[algebra]], a '''splitting field''' for a polynomial ''f'' over a field ''F'' is a [[field extension]] ''E''/''F'' with the properties that ''f'' splits completely over ''E'', but not any subfield of ''E'' containing ''F''. | ||
A splitting field for a given polynomial always exists, and is unique up to [[field isomorphism]]. | A splitting field for a given polynomial always exists, and is unique up to [[field isomorphism]]. | ||
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* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=235-237 }} | * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=235-237 }} | ||
* {{ cite book | author=P.J. McCarthy | title=Algebraic extensions of fields | publisher=[[Dover Publications]] | year=1991 | isbn=0-486-66651-4 | pages=15-16 }} | * {{ cite book | author=P.J. McCarthy | title=Algebraic extensions of fields | publisher=[[Dover Publications]] | year=1991 | isbn=0-486-66651-4 | pages=15-16 }} | ||
* {{cite book | author=I.N. Stewart | authorlink=Ian Stewart (mathematician) | title=Galois theory | publisher=Chapman and Hall | year=1973 | isbn=0-412-10800-3 | pages=86-90 }} | * {{cite book | author=I.N. Stewart | authorlink=Ian Stewart (mathematician) | title=Galois theory | publisher=Chapman and Hall | year=1973 | isbn=0-412-10800-3 | pages=86-90 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:01, 21 October 2024
In algebra, a splitting field for a polynomial f over a field F is a field extension E/F with the properties that f splits completely over E, but not any subfield of E containing F.
A splitting field for a given polynomial always exists, and is unique up to field isomorphism.
References
- A.G. Howson (1972). A handbook of terms used in algebra and analysis. Cambridge University Press, 72. ISBN 0-521-09695-2.
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 235-237. ISBN 0-201-55540-9.
- P.J. McCarthy (1991). Algebraic extensions of fields. Dover Publications, 15-16. ISBN 0-486-66651-4.
- I.N. Stewart (1973). Galois theory. Chapman and Hall, 86-90. ISBN 0-412-10800-3.