What is minimal polynomial: Definition and 42 Discussions

In field theory, a branch of mathematics, the minimal polynomial of an element α of a field extension is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1.
More formally, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E/F. 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
More specifically, Jα is the kernel of the ring homomorphism from F[x] to E which sends polynomials g to their value g(α) at the element α. Because it is the kernel of a ring homomorphism, Jα is an ideal of the polynomial ring F[x]: it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of F (which is scalar multiplication if F[x] is regarded as a vector space over F).
The zero polynomial, all of whose coefficients are 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, i.e. if the latter is not the zero ideal, 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 f(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/⟨f(x)⟩, where ⟨f(x)⟩ is the ideal of F[x] generated by f(x). Minimal polynomials are also used to define conjugate elements.

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  1. C

    I If T is diagonalizable then is restriction operator diagonalizable?

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  2. chwala

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    Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be; ##α^2=5+2\sqrt{6}##
  3. K

    I Finite fields, irreducible polynomial and minimal polynomial theorem

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  4. evinda

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  5. PsychonautQQ

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  6. PsychonautQQ

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  7. PsychonautQQ

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  8. M

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  9. K

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  10. E

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  11. M

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  12. C

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  13. C

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  14. Sudharaka

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  15. C

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  16. R

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  17. caffeinemachine

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  18. J

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  19. C

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  20. A

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  21. G

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  22. A

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  23. dexterdev

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    I have a small idea on what irreducible and primitive polynomials are in Abstract algebra. But what is minimal polynomial? -Devanand T
  24. P

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  25. A

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  26. M

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  27. 8

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  28. T

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  30. G

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  31. C

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  32. S

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  33. K

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  34. C

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  35. D

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  36. A

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  37. C

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  38. O

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  39. W

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  40. MathematicalPhysicist

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  41. V

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  42. C

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