8. (критерий Эйзенштейна) Пусть
![](data:image/gif;base64,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)
- полином в
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhAAAAAAAAAAAHR0AAB0AHQAAMwAdSAAcSB0dSAAzWh1GbDMAADMeRzQ0HTNGbjNbgEgcAFozAEhZf1l/bkhuf11/f2xGHW5GM39/XX9uSHtqRGZ3d2aIiIBbM4iIZgECAwVtICCOZGmeZhSsbAFAq4GO60EypTAms2mjgRGE1xtZfqZBiVgUIUkL5ssU0ImCoqMJcpLuAN4nIOIqDU8DRUlrFFC6J66PFFBvTxfwGjlgaggkXh0zPwssLDIiZzRCdUZiM149kHFNj5YAMUWaIQA7)
. Если существует такое простое число p, что p не делит
![](data:image/gif;base64,R0lGODlhEQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAMAAwAhAAAAAAAAB0AAAAAMwAcSAAzWh0zWh1GbDMAADNGbjNbgEgcAFozHVozAF1/f113d0huf2xGHW5GM39uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwUnICCOZLkEQTkK46EiajnEZBGnQDTnhGgADNEJl2vRjqKIMfYLHnEhADs=)
и делит остальные коэффициенты
![](data:image/gif;base64,R0lGODlhSwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACABIAAwAhQAAAAAAAB0AAAAAMx0AMgAdSAAzWgAzWx1GbDMAADIAMjMzWzNbgEgcAEgdAEgdHUceM1ozHVozAEgzM1szM1tISEZGRll/WV1/f113d0huf2xGHX9ZM25GRn9ZSHtqRH9/XX9uSHd3ZmaIiIBbM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwaDQIBwSCwaj8ikcplsBALMoRMarTafSsEQEdUKudbw0AsAGxNitHg9VC8HYjibLV8ajnXj/Rwwz/VMG3IbBVF1hUQKf0iHSQ5YRBBEAnWPVEURY3IOQ5xCngCglESWRQQAbkiSi6xMEwALSqtCCQdRDG1sqUkPACStgMDCl1VPxMJJakEAOw==)
, но
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAQABIAhAAAAAAAAB0AAB4eAB0AHQAAMwAdMh0dSAAdSAAzWh1GbDMAADIAHTIAMjNbgEgcAEgdHVozHVozAFtISF1/f0huf1luf2xGHW5GRn9/XX9uSHtqRGZ3d2aIiIBbM4iIZgU7ICCOZGmeS3CuQMOe1Suv3jwm9hXsziiokkBhNhgpXobR4yUZaQixlUskwbEOoyFLt7O+ojOsbYeYhQAAOw==)
не делит
![](data:image/gif;base64,R0lGODlhEQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACAAOAAwAhAAAAAAAAB0AAAAAMx0AMgAcSAAzWh1GbDMAADMzWzNbgEgcAEgdHUgzM1szM0ZGRll/WV1/f113d2xGHX9uSH9/XXd3ZmaIiIBbM4BuboiIZgECAwECAwECAwECAwECAwUrICCOZCkuQWCSwnisAAKXw0watjjVwFTMKBWJEMsBGoCEjQHA2JzGkQwQAgA7)
, тогда полином
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
неприводим.
Доказательство большинства из этих теорем опускается, иначе это уведёт от главной цели.
Разложение полиномов на свободные от квадратов множители. Полином
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
называется свободным от квадратов, если не найдётся полинома
![](data:image/gif;base64,R0lGODlhIQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAbAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSB00NAAzWh1GbDMAADIAHTIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwVsICCOpOiU5ImuQMGOwpvKo0FPNKmQTADggEYpMBgFDrURYQGwuFqr3Y4nikwBO+dqiWIwn4DICEwSd38jS1FEFl2wZ/gGgBgJSXOWDUDwWUgTTCI+VQGCTVcjFSWLOSgqJDEyDBA8AQGJjiQhADs=)
положительной степени, такого что
![](data:image/gif;base64,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)
. Cправедлива
Теорема. Пусть K - область с однозначным разложением на множители, характеристики нуль. И пусть
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
- примитивный полином в K[x], отличный от константы. Возьмём его однозначное разложение на множители
![](data:image/gif;base64,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)
. Его производную обозначим
![](data:image/gif;base64,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)
. Тогда НОД(
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
,
![](data:image/gif;base64,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)
)=
![](data:image/gif;base64,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)
Доказательство: Обозначим
![](data:image/gif;base64,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)
и
r(x)= НОД(
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
,
![](data:image/gif;base64,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)
). Тогда
![](data:image/gif;base64,R0lGODlhhwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQCCABAAhQAAAAAAAB0AHR0AAAAAHR0dAAAAMx0AMgAdMgAcSB0dSB00NB0zWgAzWh1GbDIAHTMAADIAMjQ0HTNdXTNGbjNbgEgcAEgdHUceM1ozAF1/f0huf1luf2xGHW5GM25GRn9/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikcslsOp/GAFQYmVqVVWR2aAlIr82t0JsYMgAUqIEpADPXyja34SbB48VOGZAJrD9OYkkhblhNB1yFT4hEewAVABNOGU8ddIpElE6XFpdXDk+eekIdAZ5DAwGQfkMPG0UBBK8kAY6MVxABoJBCr0K0BENfAIRCFkepq3dFD0XAwkMhtwCOSQ8FxkPLQx0AlkUQRX1e5KZGA9rpRnSnAOEAnUfXkMdH20Ps4NxH779FoB3uCUFnpB+UegBIcBM4ABQ4UAiJKCTisIjAEA0LVqzGRp2RiEMgVHxyJ6JAkELexRNHRE4egfD4ceNoZIumhDAlyMxUjlw7ok/Bft3zAKCdO3pHbHJZJpCoUQi8qC1RICTDsmZGQFElOmTapCELmCGhWoTRyiNWiWAggtXJrVFKyB2pWMrTMCFGm6SiMxJAxWFd6CaK4mXJzTFcdO1rpOSCkrZLBFlxLJbJlrNCKB9ZyzYQEbhHig0S4CuJQUzRSCuBOjjaFJemowqhWZVMZEOFct0lInmIoC61+NjenLRSYdTIkyvqQzt5EAA7)
и
![](data:image/gif;base64,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)
, откуда следует что
![](data:image/gif;base64,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)
. Методом от противного можно показать что
![](data:image/gif;base64,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)
не делит
r(x). Предположим что
![](data:image/gif;base64,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)
. Тогда
![](data:image/gif;base64,R0lGODlhZwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQBiABAAhQAAAAAAAB0AHR0AAAAAHR0dAAAAMx0AMgAdMgAcSB0dSB0zWgAzWh1GbDIAHTMAADIAMjMeRzQ0HTMzWzNdXTNGbjNbgEgcAEgdHUceM1ozAE4zAEZGbl1/f0huf1luf2xGHW5GM25GRn9/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8jkMKBsOp/ES4AJNUKSV+I0MVwAKslJVWioZocXxrhYVgqMXIAm0F4jL8R68o22D89KB0VxABYAFGFrhmOCQml+GlVqQiBxIAGTfkYRQw2SjkUkA0xzekUeQycBBEtDJINVYkajhnRGDkWqrEJUACSNAJlDDgWOnkamQmrCAA9EhEPOY8SgbEjLRdJ4Rgh9RiDJAAPHRA/k0E2yuUXkZEei7ULmAI9EkUN8yEfb2UOVmkjyUQpH74i2JpnqnEgWIpiRB4sAoFOijsiZe9aMNGTWzFC9IQqEaCAIABeRE0JCFgEmEWBKkeFMQmnEj9IUTEUysGvlKAA5SRCwoFRcMiUJRl5ofPpDIwxDEp1EZCoB9A+gUydSp3oD+TSUAFRJIAZ9MvQVFIFIxBLlssWlRSKXAkyEMkcuFEBWAPZyG4svlCAAOw==)
, откуда можно заключить что
![](data:image/gif;base64,R0lGODlh0QAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwDMABIAhQAAAAAAAB0AHR4AHh0AAAAAHR0dAAAAMx0AMgAdMgAcSB0dSB00NB0zWgAzWh1GbDIAHTMAADMAHTIAMjMAMzMeRzQ0HTMzWzNdXTNGbjNbgEgcAEgdAEgdHUceM1ozAFszAE4zAEZGbl1/f113d0huf0Rqe1luf2xGHW5GM25GRn9/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikcslsOp/PzQFKrVqv2Kx2y+16v2BiIEx2jpWesnq4CZzBk6JbMWwAMskLdU6OiwN0QnZ4WyhTRX5rQxRFGw5Xh0oCRoEfAZFbG1mYSJNFlZdEbXwAaU2ORZxHnk2qRgOiigifQxoAGHlUj2uzRIEAtriZiE0sYL0AqFQfD0+7ACiBKAHPmdVPzM5D0ULT10aAVR9PKN9F2U67mkQsBGOWrhBFLwEFQ2/Gvk96RqNH1+3ehTIijwg9e0Lw0criztZAEUQglJBTYCK9X8iERAjQzFbBIQfvDckn5BsEA0I2NDPoCsCjaxH0FYnphCaAXymRnEzyouXL/5nbcGqksjPZEVcooMEkQmDIlJ5IfhKhua5IAjZHkRBYOZWrUCX8sAoZZ8Tc1ZxGWm41EsGrlhdFuHZTe8RmVbhC1MrV2EzZELJCWG1rWXVqUCwII3zz+5eI4LlICg+xyc3L47yR666MtM5QZrZokzyL5CqFS7a2hHzN9YczJnNEHkkhUvr0zNQ3tSQCHAnqEQt1hzxDCKD2twi2GANYMLblR5BCmBfJuNpJJE+mjBqR/iFSdgDPi0gnQl0Ld+8RjzRjbnpIL8AM0iMZ7z50NzfUimQHnJANx20LNRFWSm6IVsQ0bsAGAH8AvNEGV0nJhAV++sW1TX79xTaEO49wBY6CHENw8F83bFTTQRLfhZdEItBUd0V2yqFYhIpIsFjZFSc2QeMRLA6R4xASOJGIZPRV6BgTyAXIxIBNXFOkJEjiphoWJBnxHQAsDDCREjYxAcsSSd4TDilN9EjMheF4ASMpZBJxpRBmEmFjm1BYkqaRRTCShJ4zFWgEn0cAuociRjBphCUuIoEooRMy+kQQADs=)
. Отсюда после сокращений
![](data:image/gif;base64,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)
. Cтало быть
![](data:image/gif;base64,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)
потому что НОД(
![](data:image/gif;base64,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)
)=1. Из этого можно заключить что
![](data:image/gif;base64,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)
. Очевидное противоречие.
Из теоремы легко выводятся два следствия.
Следствие1. Простые корни полинома не являются корнями его производной.
Cледствие2. Пусть K – поле,
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
- неприводимый полином в K[x], который делит
![](data:image/gif;base64,R0lGODlhTwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgBLAA8AhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSB0dSAAzWgA0SB1GbDMAADIAHTIAMjMeRzQ0HTNbgEgcAEgdHVozAEYzRkZGM1tISF1/f0huf113d2xGHW5GM39/XX9uSHtqRGZ3d3d3Zm6AbmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwbsQIBwSCwaj8ikcslEOpTPpjJAFW6oB4AEOw1kk6QCUyAFByJDw1BS/jql6vLR7dYmmpu6cVG+Gz0DVAFiR2wADGhEhk11DFR8DUYBBENeaUYkUgIWhEZ+QpOVXxtFd5lhSHefQgxyRaGFR6pFdXFCnUUDfEUMu0OOgoNHFFq4a6tCukV5RJlWxmtHra6saIuKsUZ6xLdHHADIh0YTTZ0MehLI38jMWkPhQ85HtkTQRdcAA/dF8tqsVJ24VVoTwNcGeACCGRP0JdiudAOFbDGop0gkahiT4FNSsQiZJYgyNkG4rEkUkSi3BFjCJQgAOw==)
. Тогда
![](data:image/gif;base64,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)
если и только если
![](data:image/gif;base64,R0lGODlhTAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwBGAA8AhQAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjMeRzQ0HTMzWzNbgEgcAEgdHVozAFtISEZGbl1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwbzQIBwSCw+isQjUihZIg1GJ0ApXUKrg6qWUrxKs9pkGHBYNsOTIbVaHlvGwgW8CBm+4fKhIJAOeBsaRQEEgSEBCQAfbURnRA4BAQxIgEOGBEMBQ4pFDQVCXEJeRBwAHHlCDmNyIVZLcqeoRKxEkhyiRAJIDpKMc7ZIHwK8jsNFYAC3Q6COcIitSMvManaVyUIRumm9SHdFIckdALCo2gAKQhaiDUuS5+FCi06gFuNC63DxkJlIwxwBp/tKmXEEyZqQYQEpRBpCakiFKve0rJkzaQyVD1o+DAgkxUG5bXM0cnTiER0kZ1ImUlw5haXLMI1eagkCADs=)
.
Пусть
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
- примитивный полином, определённый на области с однозначным разложением на множители K,
![](data:image/gif;base64,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)
. Пусть
![](data:image/gif;base64,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)
. Для
![](data:image/gif;base64,R0lGODlhNwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAA0AAsAhAAAAAAAAAAAHR4AHgAAMwAcSAAzWh0zWh1GbDMAADIAMjNbgEgcAEgdHUceM1ozAF1dRl1/f0Rqe0huf113d2xGHX9/XX9uSHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwV1ICAWYmmeaKquZxKQbGmcFRED1XyPcRXoO1MgKKrAUo8hsZRELHkoRmCxclFRypg0Cz0FuClH6ntLlI4A9KmZusTAJgbTZLzBAU6WL28SiBJAXXonDUSBAF82a2pBChBPIhuKAAMlLgF+Xl+bYHc+nF+HWyYhADs=)
положим
![](data:image/gif;base64,R0lGODlhaAAZAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABkABIAhQAAAAAAAB0AAB0AHQAAHR4AHh0AMgAAMwAcSB0dSAAzWgA0SB0zWh1GbDMAADIAHTIAMjMzWzNbgEgcAEgdHUceM1ozAFozHVszM0YzRkZGM11dRltISF1/f0huf113d0Rqe2xGHX9ZSH9/XX9uSHtqRG5/WXd3ZmZ3d2Z/ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb1QIBwSCwaj8ikcslsMkOBaAMADTiXlmvkylVakV+hoYu0IMoHsvqYFgqQZ2F8TXRItPT8fDgROvJGfYCDS4JEJAMabYRDCkdhRFuBUYwOkA5TRQEEVxZRn1GORQVJmUt/clign6JKi0JZE6+De0UXR5J8Q1mMDEZ2Q8ATtYBjdJwADq2MRMZDi6iqqwHLjUemkZoBs0IYj9NHHkXYhm7EaxPVAOWuQ6SEEEXln0OgU7xrmEUkzkW5QhMoGblDZgW1IuqYFYlGBJtCNfiaeMszq5+/h0wMEsQIJ1mAc2R8DQqB7AokjtmeoFzJsgLLlwotEBDHMQgAOw==)
,
![](data:image/gif;base64,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)
. Тогда
![](data:image/gif;base64,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)
называется разложением полинома
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
на свободные от квадратов множители.
Замечание. Некоторые из полиномов
![](data:image/gif;base64,R0lGODlhJAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAfABAAhQAAAAAAAB4AHgAAHR0AAAAAMx0AMgAdSAAcSB0dSAAzWgAzWx1GbDMAADIAHTMAHTMAMzIAMjQ0HTMzWzNZfzNbgEgcAEgdAFozAFszM0YzRltISF1/f113d0Rqe0huf2xGHW5GM3d3XX9/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZ1QIBwSCwaj8ikcrksLAXMaJEkFWIYUZCC2AgEjo7iKTAYfoUG4vZ4ch63Cy4xjXQbCVg5EXEE2YsWRg1YG0N5ABpDf0MhAGtDDUKBABhIYWJCCUV0XQF/fFZFZwAXAYePTA9REVVCUEqRShNHEEm1SmOHrUJBADs=)
могут быть единицей,
![](data:image/gif;base64,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)
- произведение всех линейных множителей, cоответствующим простым корням,
![](data:image/gif;base64,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)
- произведение всех линейных множителей, cоответствующим двойным корням и т.д.
Так как r(x)= НОД(
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
,
![](data:image/gif;base64,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)
)=
![](data:image/gif;base64,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)
(здесь без
![](data:image/gif;base64,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)
).
Наибольший свободный от квадратов делитель полинома
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
равен
![](data:image/gif;base64,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)
.
Cледовательно,
НОД(
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,
![](data:image/gif;base64,R0lGODlhHwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAZAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNdXTNGbjNbgEgcAEceM1ozAF1/f0huf1l/bmxGHW5GM39/XX9uSGaIiIBbM4iIZgECAwVcICCOY0OegImSXbGewiuq8mjAo1KfyRj8gMWpExj4DjaSrvXq9UiMkVDkWhF0J4YOM8JUUZJVFBAefU+OJ1R0m4wsQ8DmnQQUR5RT4CkJYAFcL1M7hDE7YzU0hCEAOw==)
)=
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.
Поэтому
![](data:image/gif;base64,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)
. Повторяя процесс с
![](data:image/gif;base64,R0lGODlhIAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAaAA4AhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSB0dSAAyMh00NAAzWh0zWh1GbDIAHTMAADIAMjQ0HTNdXTNbgEgcAFozAEhOM11/f0huf2xGHW5bNW5GM39/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZpQIBwSIQQjwAjEllYHgUYp1ApHRqkleoxMxw0QQGAAxkgDAMHK1EjBjSRXO7xMREqhpk3crB8CLNEekh1cw1JR4JEEX1CC0djSIYIABtDHX9LhkIZAY4AFGFEnksOUVqHVQKnfqdaVABBADs=)
вместо
![](data:image/gif;base64,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)
мы можем вычислить
![](data:image/gif;base64,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)
как первый свободный от квадратов сомножитель
![](data:image/gif;base64,R0lGODlhIAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAaAA4AhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSB0dSAAyMh00NAAzWh0zWh1GbDIAHTMAADIAMjQ0HTNdXTNbgEgcAFozAEhOM11/f0huf2xGHW5bNW5GM39/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZpQIBwSIQQjwAjEllYHgUYp1ApHRqkleoxMxw0QQGAAxkgDAMHK1EjBjSRXO7xMREqhpk3crB8CLNEekh1cw1JR4JEEX1CC0djSIYIABtDHX9LhkIZAY4AFGFEnksOUVqHVQKnfqdaVABBADs=)
, и в конце можно получить все свободные от квадратов сомножители
![](data:image/gif;base64,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)
. Таким образом получен алгоритм, известный под названием PSQFF(
Polynomial
Square
Free
Factorization).
Вход:
![](data:image/gif;base64,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)
- примитивный полином, определённый на области с однозначным разложением на множители K,
![](data:image/gif;base64,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)
, char(K)=0.
Выход: полиномы
![](data:image/gif;base64,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)
и вышеопределённое число e, определяющие разложение
![](data:image/gif;base64,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)
на свободные от квадратов множители.
На условном языке программирования алгоритм выглядит примерно так:
BEGIN // первоначальная инициализация
![](data:image/gif;base64,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)
![](data:image/gif;base64,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)
j:=1
label:
IF
![](data:image/gif;base64,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)
THEN // выход?
BEGIN
e:=j
![](data:image/gif;base64,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)
EXIT
END
v(x):=
![](data:image/gif;base64,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)
// вычисляем
![](data:image/gif;base64,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)
![](data:image/gif;base64,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)
![](data:image/gif;base64,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)
// обновляем
![](data:image/gif;base64,R0lGODlhTwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwBJAA4AhQAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh0zWh1GbDIAHTMAADIAMjMeRzQ0HTMzWzNdXTNGbjNbgEgcAEceM1ozAF1/f0huf0Rqe1l/bmxGHW5GM39/XX9uSHtqRG5/WWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwbTQIBwOHQQjwAjcslsIpVLKJNUcAoF1my26sQ2pU6DdvzUiolewIKcILsv7GFgDmAcSYGB/DA8EwNueUNwdUiCQgF8AH5qAFRMbW1EDU0Xc5dzkpNrQhEAXEccAKKTQ3ZCoEcEnJtuSxeao6lEBEgNa6SyTBVLlK5LfB9Ds0QUR768w0wQsUK+TJiXTHwPyksQtkJiFkMZdwDCQt7av0sBqadHawgAHX2IekKEcpIVAay5f7+KRPgB9YCE5DuSrpxBJAzGHUzT5NlBKwx7ZQHzsKKWIAA7)
INCR(j)
GOTO label
END
Основные факты о конечных полях. Из определения поля видно, что каждое поле – область целостности, обратное утверждение в общем случае неверно. Но имеет место следующее утверждение:
Каждая конечная область целостности – поле.
Если взять два неравных элемента a,b из конечной области целостности K , то для всех ненулевых элементов
![](data:image/gif;base64,R0lGODlhKQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAlAAwAhQAAAAAAAB0AAB0AHQAAHR0AMwAAMwAdSAAcSAA0SAAzWh1GbDMAADMeRzNbgEgcAEgdAEgdHVozAEYzRkZGM11/f0Rqe113d2xGHX9uSHd3XXtqRH9/XXd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZjQIBwSCwaj8ikMhloDpuHpdQYIBam2CFmKMkSBc2AAflYABjJh9MYlQ4AY6QgOcEGCEl0PtyMGxNwSH5HDVN6EAhHCksWSyB+DIlEGHEYbUSPUotDApJDEGtHIHyDo1AApqFBADs=)
по правилу сокращения
![](data:image/gif;base64,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)
. Поэтому
сК=К и найдется такой
![](data:image/gif;base64,R0lGODlhLAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAoAAwAhQAAAAAAAB0AAB0AHQAAHR0AMwAAMx0dNAAcSAAdSAAzWgAzWwA0SB1GbDMAADQdNDMeRzNbgEgdHUgcAEceM1ozAEYzRkZGM0hZf1tuSF1/f0huf113d2xGHW5bNX9ZM39uSH9/XXd3XXtqRG5/WWZ3d3d3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZzQIBwSCwaj8ikMKBsdgLQBuDJTEqbykBkWFBKsE3EMAE+CqABQ3IidCQrAbHiSMaCBhc10rEQCo4WZQABBGtYDmhQekYVABOLRZBzRhBlbo5iRpMAf0gbYIuXRI1CB0qQSGxEAplDZ2SiRiiJqINQQ4lXQQA7)
, что
![](data:image/gif;base64,R0lGODlhLAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAoAAwAhQAAAAAAAB0AAAAAHQAAMx0dNAAcSAAdSAAzWgAzWx1GbDMAADQdNDMeRzNbgEgcAEgdHUgdAEceM1ozAEhZf1tuSEhuf11/f113d2xGHW5bNX9ZM39/XXtqRHd3XX9uSG5/WXd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZrQIBwSCwaj8gkkhQwKJ/OJzEjRS6aVaIle4xmG1xjxit8BAJFtCbcJQqMi4TwfZyc72f51Ls4IoYSbEZeBEaFQ3+CQ2OIRRNDHwIOSXh3bQAZhxkHABOJBQyKRBNeEWeQAmgAC5yiVwEDFkEAOw==)
, что и означает наличие у каждого ненулевого элемента конечной области целостности мультипликативного обратного элемента, что и подтверждает что K- поле.
Так как ненулевые элементы любого конечного поля из q элементов образуют абелеву группу порядка q-1 относительно умножения, то справедлива