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[[Fasciculus:Polynomial roots multiplicity.svg|thumb|Polynomium <math>x^3 + 2x^2 - 7x + 4</math> tres radices (-4, 1, 1) habet.]]
'''Theorema fundamentale algebrae''' dicit omne [[polynomium]] unius variabilis, gradus ''n,'' ''n'' radices habere, vel, quod idem est, ''n'' [[in factores resolutio|factores]] lineares, in [[corpus (mathematica)|corpore]] "pleno" sicut est corpus [[numeri complexi|numerorum complexorum]].
Radix quaedam potest plus quam unus factor polynomium repraesentare; tunc dicimus radicem duplicem vel multiplicem esse. Exempli gratia, <math>f(x) = x^3 + 2x^1 - 7 x + 4 = (x + 4)(x - 1)^2</math> Aequatio <math>f(x) = 0</math> habet solutiones ''x = -4'' et ''x = -1,'' hic autem bis in solutione init quod ''(x - 1)'' est bis factor. Duae ergo trium radicum huius polynomii eaedem sunt.
Corpus [[numerus rationalis|numerorum rationalium]] non est plenum: polynomium <math>x^2 - 2</math> nullam radicem hoc in corpore habet. Nec corpus [[numerus realis|realium]]: <math>x^2 + 4</math> nullam radicem realem habet. Corpus [[numerus algebraicus|numerorum algebraicorum]] autem plenum est, per definitionem.
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Corpus numerorum complexorum non modo plenum [[algebra|algebraicum]] verum etiam plenum [[spatium]] [[topologia|topologicum]] est.
[[Demonstratio mathematica|Demonstratio]] theorematis non algebraica sed [[topologia algebraica|topologica]] est.
==Bibliographia==▼
▲== Bibliographia ==
*Shipman, Joseph. [[2007]]. Improving the Fundamental Theorem of Algebra. ''Mathematical Intelligencer'' 29(4): 9–14, ISSN 0343-6993. doi:10.1007/BF02986170.▼
*
▲* Shipman, Joseph. [[2007]]. Improving the Fundamental Theorem of Algebra. ''Mathematical Intelligencer'' 29(4): 9–14, ISSN 0343-6993. doi:10.1007/BF02986170.
*Smith, David Eugene. [[1959]] ''A Source Book in Mathematics.'' Dover. ISBN 0-486-64690-4.▼
* Smale, Steve. [[1981]]. The Fundamental Theorem of Algebra and Complexity Theory. ''Bulletin of the American Mathematical Society'' series nova, 4(1).
▲* Smith, David Eugene. [[1959]] ''A Source Book in Mathematics.'' Dover. ISBN 0-486-64690-4.
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