minimal polynomial
Let K/F be a field extension and κ∈K be algebraic over F. The minimal polynomial for κ over F is a monic polynomial m(x)∈F[x] such that m(κ)=0 and, for any other polynomial f(x)∈F[x] with f(κ)=0, m divides f. Note that, for any element κ that is algebraic over F, a minimal polynomial exists (http://planetmath.org/ExistenceOfTheMinimalPolynomial); moreover, because of the monic condition, it exists uniquely.
Given κ∈K, a polynomial m is the minimal polynomial of κ if and only if m(κ)=0 and m is both monic and irreducible (http://planetmath.org/IrreduciblePolynomial).
Title | minimal polynomial |
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Canonical name | MinimalPolynomial |
Date of creation | 2013-03-22 13:20:11 |
Last modified on | 2013-03-22 13:20:11 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 13 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 11C08 |
Classification | msc 11R04 |
Classification | msc 12F05 |
Classification | msc 12E05 |
Related topic | DegreeOfAnAlgebraicNumber |