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- Minimal_polynomial_(field_theory) abstract "In field theory, a branch of mathematics, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E. The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let Jα be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in Jα. The set Jα is so named because it is an ideal of F[x]. The zero polynomial, whose every coefficient is 0, is in every Jα since 0αi = 0 for all α and i. This makes the zero polynomial useless for classifying different values of α into types, so it is excepted. If there are any non-zero polynomials in Jα, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in Jα. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of Jα, then α is called a transcendental element over F and has no minimal polynomial with respect to E/F.Minimal polynomials are useful for constructing and analyzing field extensions. When α is algebraic with minimal polynomial a(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/⟨a(x)⟩, where ⟨a(x)⟩ is the ideal of F[x] generated by a(x). Minimal polynomials are also used to define conjugate elements.".
- Minimal_polynomial_(field_theory) wikiPageID "9667106".
- Minimal_polynomial_(field_theory) wikiPageRevisionID "597843450".
- Minimal_polynomial_(field_theory) hasPhotoCollection Minimal_polynomial_(field_theory).
- Minimal_polynomial_(field_theory) title "Algebraic Number Minimal Polynomial".
- Minimal_polynomial_(field_theory) title "Minimal polynomial".
- Minimal_polynomial_(field_theory) urlname "AlgebraicNumberMinimalPolynomial".
- Minimal_polynomial_(field_theory) urlname "MinimalPolynomial".
- Minimal_polynomial_(field_theory) subject Category:Field_theory.
- Minimal_polynomial_(field_theory) subject Category:Polynomials.
- Minimal_polynomial_(field_theory) type Abstraction100002137.
- Minimal_polynomial_(field_theory) type Function113783816.
- Minimal_polynomial_(field_theory) type MathematicalRelation113783581.
- Minimal_polynomial_(field_theory) type Polynomial105861855.
- Minimal_polynomial_(field_theory) type Polynomials.
- Minimal_polynomial_(field_theory) type Relation100031921.
- Minimal_polynomial_(field_theory) comment "In field theory, a branch of mathematics, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E. The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let Jα be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in Jα.".
- Minimal_polynomial_(field_theory) label "Minimal polynomial (field theory)".
- Minimal_polynomial_(field_theory) label "Minimale polynoom".
- Minimal_polynomial_(field_theory) label "Polynôme minimal (théorie des corps)".
- Minimal_polynomial_(field_theory) label "Минимальный многочлен алгебраического элемента".
- Minimal_polynomial_(field_theory) sameAs Polynôme_minimal_(théorie_des_corps).
- Minimal_polynomial_(field_theory) sameAs Minimale_polynoom.
- Minimal_polynomial_(field_theory) sameAs m.02pnlcl.
- Minimal_polynomial_(field_theory) sameAs Q2242730.
- Minimal_polynomial_(field_theory) sameAs Q2242730.
- Minimal_polynomial_(field_theory) sameAs Minimal_polynomial_(field_theory).
- Minimal_polynomial_(field_theory) wasDerivedFrom Minimal_polynomial_(field_theory)?oldid=597843450.
- Minimal_polynomial_(field_theory) isPrimaryTopicOf Minimal_polynomial_(field_theory).