| Revision as of 22:37, 28 August 2005 editDysprosia (talk | contribs)28,388 editsm Reverted edits by 24.193.245.239 to last version by MarSch← 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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| {{disam}} | |||
| 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. | |||
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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: