| Revision as of 03:53, 14 October 2005 editKusma (talk | contribs)Autopatrolled, Administrators59,515 edits the point of my edit was the 4 in the exponent, which is correct iff n=4← Previous edit | Latest revision as of 20:57, 11 October 2022 edit undo173 Ascension 257 (talk | contribs)11 editsm The minimal polynomial of 2cos(2π/n) is related to the minimal polynomial in field theory. I added it as a subcategory of the said topic. | ||
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| '''Minimal polynomial''' can mean: | |||
| The '''minimal polynomial''' of an ''n''-by-''n'' ] ''A'' over a ] '''F''' is the ] ''p''(''x'') with leading coefficient 1 over '''F''' of least degree such that ''p''(''A'')=0. Any other polynomial ''q'' with ''q''(''A'') = 0 is a (polynomial) multiple of ''p''. | |||
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| The following three statements are equivalent: | |||
| #λ∈'''F''' is a root of ''p''(''x''), | |||
| #λ is a root of the ] of ''A'', | |||
| #λ is an ] of ''A''. | |||
| The multiplicity of a root λ of ''p''(''x'') is the ''geometric multiplicity'' of λ and is the size of the largest ] corresponding to λ and the dimension of the corresponding eigenspace. | |||
| The minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix <math>4I</math>, it has characteristic polynomial <math>(x-4)^n</math>. However, the minimal polynomial is <math>x-4</math>, since <math>4I-4I=0</math> as desired, so they are different for <math>n\ge 2</math>. That the minimal polynomial always divides the characteristic polynomial is a consequence of the ]. | |||
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| In ], given a ] ''E''/''F'' and an element α of ''E'' which is ] over ''F'', the '''minimal polynomial''' of α is the monic polynomial ''p'', with coefficients in ''F'', of least degree such that ''p''(α) = 0. The minimal polynomial is irreducible, and any other non-zero polynomial ''f'' with ''f''(α) = 0 is a multiple of ''p''. | |||
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Latest revision as of 20:57, 11 October 2022
Minimal polynomial can mean:
Topics referred to by the same term
This disambiguation page lists articles associated with the title Minimal polynomial.If an internal link led you here, you may wish to change the link to point directly to the intended article. Category: