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 not not any subfield of ''E'' containing ''F''. | 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 not any subfield of ''E'' containing ''F''. | ||
Revision as of 12:10, 21 December 2008
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 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.