Введём для
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAfAAwAhQAAAAAAAB0AABwcHAAAHQAAMwAcSAAdSB0dSAAzWgAzWx1GbDMAADMAHTIAHTIAMjNGbjNbgEgcAEgdAEgdHVozAFszM0hbSFl/WV1/f0huf2xGHX9/XX9uSHd3XXtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZfQIBwSCwaj8ikcihZEjdOACNgiAo3gcQSaiUGDsqqUBCIACaBRbIjKBy5wwegEZ2qjWLACGAZao4fbUlwABUDQwhHX055DEOEQwpRkEMOXUYVeXcAAooBn6BGUwEEAEEAOw==)
функцию Мёбиуса
![](data:image/gif;base64,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)
следующим образом:
![](data:image/gif;base64,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)
если
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAfAAwAhQAAAAAAAB0AABwcHAAAHQAcSB0dSAAdSAAzWwAzWh1GbB1IbjMAADMAHTIAHTIAMjNGbjNbgEgcAEgdAEgdHVszAFozAFszM0hbSFl/WUhuf11/f2xGHW5IHX9/XX9uSHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZQQIBwSCwaj8ikcihZOo2MQOFJFXKq2KlQEIgAJgEFdnglPgCNZGDNDhy1gBHgMtSMrUTLYGi4D+EMZEcVbWwIRWVGDn5CFnALQwKMUQEEAEEAOw==)
если
![](data:image/gif;base64,R0lGODlhKAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAjABEAhQAAAAAAAB0AAB4eAB0AHRwcHAAAMwAdMh0dSAAdSB0zWgAzWwAzWh1GbDIAHTMAHTMAADIAMjMzWzNGbjNZfzNbgEgdAEgcAEgdHVozAFszAFszM0hbSFl/WV1/f0huf1luf2xGHX9uSHtqRH9/XXd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZyQIBwSCwaj8ikMBRoKJ9KEWAErR4zAQNSYk1mAKiuWBhYbLuCAAWQHQNE6WGg6BgIL1VuMQJ4GMNETm4MQxhJBHlFGQVDCER8Ql9uABCCAJYAjgAZWolEmgAORQGkk0tDcAmFk3pPVKZQGqSqY62wtwBBADs=)
для некоторого простого p и некоторого
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAAwAhAAAAAAAAB0AAAAAHR4AHgAAMwAcSAAdSAAzWgAzWx1GbB1IbjMAAEgcAEgdAFozAF1/f0Rqe113d0huf25IHWxGHX9ZM39/XX9uSHd3XXtqRHd3ZmZ3d2aIiIBbM4iIZgVNICCOZGmeaKqKxjoqLhO0rhjUdG0faZWLjcBNJSigfozRzyQ7jhqjR6qoyg1EjMQpwFP5RkKjiaID0DxiAKFceuQcwlNFSBdqSbLANQQAOw==)
если n раскладывается в произведение r различных простых чисел
Если n делится на квадрат простого числа, то
![](data:image/gif;base64,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)
; для простого числа
p ![](data:image/gif;base64,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)
. Также m и n – взаимно простые числа, то
![](data:image/gif;base64,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)
, то есть
![](data:image/gif;base64,R0lGODlhEAARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAMAAwAhAAAAAAAAB0AHR0AAB4eAAAAMwAcSAAzWwAzWh1GbDMeRzMzWzNbgEgcAEgdAEhuf11/f2xGHW5IHW5GM39uSHtqRGp7VX9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwUrICCOZCkKY2RGhlkmI1rKAFuqJEw2OkCPg1HANCEEAgdAoOBqAhBOgCMaAgA7)
- мультипликативная функция. А для мультипликативных функций верна теорема
Если f – мультипликативная функция, а функция F определена соотношением
![](data:image/gif;base64,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)
, то
F – также мультипликативная функция.
Доказательство: Пусть числа m и n – взаимно простые. Тогда каждый делитель d числа
![](data:image/gif;base64,R0lGODlhIQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAcAAgAhQAAAAAAAB0AAAAAHRwcHAAAMwAdMh0dSQAdSB0dSAAyMgAzWgAzWx1GbB1IbjMAADIAHTMAHTIAMjQeSDMzSDNGbjNbgEgdAEgdHUgcAFozAFszAFszM0hbSFtuSFl/WV1/f0huf2xGHW5GM39/XXtqRH9uSG5/WXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZVQIBwSCwaj4BHwAIQFIROJHQJuAQaQgjgAJBsu1KhN0JUASgA8yQdZnOGIYBmINQ85UUBg6ghDBNCGUMPQ15hhEIiRioIQmZtRFpCDkNMQgFtlFBCQQA7)
может быть представлен в виде произведения взаимно простых
![](data:image/gif;base64,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)
, таких что
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,R0lGODlhJwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAhABAAhQAAAAAAAB0AABwcHAAAMx0dNAAcSAAdSB0dSAAzWgAzWx1GbB1IbjMAADMAHTIAHTIAMjQdNDMeRzMzWzNGbjNbgEgdHUgdAEgcAEceM1ozAFszAFozHVszM0hbSFtISEhZf1l/WVlsRkhuf11/f2xGHW5IHWxZNX9ZM25GRn9/XX9uSHd3XXtqRG5/WXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZuQIBwSCwaj0hiIMlsDhfNifNomVqvQ6lVEzgAEljASrAUlgENhVAw1RIhAIcQigkjwcIqgEDER4saA0Z+bEQNAVBNDQxPABtDBUhwRW5CCEMPABQAZF4NRxxNJkQCXnZCladCFapNfkSpdgGzp0EAOw==)
. Поэтому
![](data:image/gif;base64,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)
Теперь ещё небольшой факт:
Если
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAfAAwAhQAAAAAAAB0AABwcHAAAHQAcSB0dSAAdSAAzWwAzWh1GbB1IbjMAADMAHTIAHTIAMjMAMzNGbjNbgEgcAEgdHUgdAFozAFszM0hbSFl/WUhuf11/f2xGHW5IHX9ZM39/XX9uSHtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZZQIBwSCwaj8ikcjhZOo2MQOFJFXKqRwpyKhQEJIBKQLEkG6/EB6CRDLjfgSMXUAJchpqkJYkWWgZDBkqCSXMMQ31EHXBvCEWJRA5KjnpzC0MCSRBKUQEEAEEAOw==)
, то
![](data:image/gif;base64,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)
.
Доказательство: Функция
![](data:image/gif;base64,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)
является мультипликативной, если
e=0 ![](data:image/gif;base64,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)
и в то же время
![](data:image/gif;base64,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)
, если
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAAwAhAAAAAAAAB0AAAAAHR4AHgAAMwAcSAAdSAAzWgAzWx1GbB1IbjMAAEgcAEgdAFozAF1/f0Rqe113d0huf25IHWxGHX9/XX9uSHd3XXtqRHd3ZmZ3d2aIiIBbM4iIZgECAwVNICCOZGmeaKqKxjoqLhO0rhjUdG0faZWLjcBNJSigfozRzyQ7jhqjR6qoyg1EjMQpwFP5RkKjiaID0DpiAKFceuQcwhNFSBdqSbLANQQAOw==)
. Если n делится на простое число, то
![](data:image/gif;base64,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)
, из этого всего и следует это утверждение.
Формула обращения Мёбиуса. Для любой функции f, определённой на множестве натуральных чисел (не обязательно мультипликативной), если
![](data:image/gif;base64,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)
для каждого
![](data:image/gif;base64,R0lGODlhJQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAhAAwAhQAAAAAAAB0AHR0AABwcHAAAMwAcSAAdSB0dSAAzWgAzWx1GbDMAADMAHTIAHTIAMjNZfzNGbjNbgEgcAEgdAEgdHVozAFszM0hbSFl/WV1/f0huf2xGHX9/XX9uSHd3XXtqRGZ3d3d3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZsQIBwSCwaj8ikssgJBJZEDlRYEDKmwmZiKRkasMPAASkglgGDQJcSWCQ9gyqRJLcOH4DGlNEu1gFXACQAF0MbRyBxR3REA0IWBEMIR2JKZ0JngQBSRgpTXUJfRQ5gRqKabkKORk6tTkdNT0JBADs=)
, то
![](data:image/gif;base64,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)
.
Доказательство: Положим
![](data:image/gif;base64,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)
. Тогда суммы очевидно равны. По определению
F ![](data:image/gif;base64,R0lGODlhYQFIAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgBdAUMAhgAAAAAAAB0AABwcHB0AHQAAHR4AHh4eAAAAMx0AMh0AMxUGKB0dNBQLMxUMMx0dSAAcSAAdSBUVQBUVQQAkMgAzWh0zWgAzWwA0SAszbR1GbB1MfxVAgDMAADMAHTIAHTIAMjMAMzQdNDMeRzQeSDQsNDMzWzNBSTNGbjNZfzNbgEgcAEgdHUkdHUgdAEceM0geNFszAFozAFozHV4oC10oBlszM0gzM0hOM0hbSEZfRkZGRl1dRkhZf1l/WVlsRlV7VUhuW11/f0huf113d0Rqe1V3d0ZgbmxGHW5IHW5GM2xZNW5GRn9ZSH9/XX9uSHd3XW5/WWp7VXtqRH9/QGZ3d3d3ZmaIiIBAFYBbM4BdM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwf/gACCg4SFhoeIiYqLjI2Oj5CRkpOUlZaXmJmam5ydnp+GHQGjAaCVMaQBFaaEoqSssI6opKuxtre4gya5vL2+m7u/wsPErMGwx8XKy8zDyc3Q0Z7P0tXWv9Sf2dfc3d6C29/i3eHj5ueN5cDo7O2+6u7xvPDy9eS29Pb6+4n5/P+Y/AEc+A4fwYMIAQhMyBDRwoYQ18V6GLFiM4oWE2IshoBRh4zYDII8J+NRjHgbRwJM6QuJSmsjRFokMe7JpBntWL7URzPhg52wYk4E2g1FpSHodBKN19OdgFIAVkAtlGUpKKHIQDath1TDIazjlFplt5VbuFoAWCB6MbYTWFNi/9tusolIxgAVAOgewnkurlxxZa9lQNTBqyDDf029BeV3XM9UkCEj7oUl0U9BHwAYNVRjUuTPASZ3eoYEdGRvpiUvSp1K9LmmoyAk+igJBCLbhz7LBsDhkMtBAiIgqtFbUuzZk0IgUs7oOGHXhUSJZm4ogGzqhhY3cn6Itq7c1pFLwk6IPOGpgswLCq8e0aiOiDY/gn/oCQGkhgoU8lrZUmdK7yUinyP0GfKEAUV8Z0gWAda3iAcAVEVIgYX8BoABnTR4yICJaGgIh4xQSMiBCSbiQYLCCSLiIBZiiEhZT+mHCW6KJGAIErs1E2MmNCZiIyM7HhJOSYb0aIhagyjwVf8lQS4STpOWtEeIkojEMN0jVBoSmCgyNmIfIbIRuYiFhEDXXyX/VcLlIwSwOIiYilhYDpclNpKFiHAuSQhamqxpiZ9euihImI0kkQiFJylSFp+DBAaAC6M8kuMKmB0iVWiD/DhIZoJcxgyk6CWC4yArGMYpIZcWoukioDoCKoWnFmLdEoasKoh2kLSqiDq6NpKjC4JAaAioiGU5yKuECFsIKrJdUIixjPSqCKWC2IeUiIUBQGYHeAnC4CuESFgJmQDMAhmjw0bKCGJtqhjdIDkCwC0AEmyHKSN5uhvdKk8I0O0g3g2yQHWnKSLtJAcjAmy1BlwbilfbpnAIo9jyK4D/xK0cMnBdjrQ7qLYUroCuvITIgBZ0xeR7iMcA7IZEgdQCkGfA+I7c3b+CvFyIyIOw9W4huEZi8q6NqHyIoC3ru+cgiNEsCM1JUOgCWjA8rCUmkw5CoYwd8Jnt04M0QRV4oN2IC7mI0ZcF11bTE3Sth1Codnf1/Kp1IVx7jXIhcg8ibsZXWxIzAB4XWFoA2BLSpS2faVLqICwX6CHAjyDZyKiGFDi0IALQDfSyn9ksSAuMwEN6I4MLWmASk5NsSIoTEpIo5547gnMi/XrFoGy7xAoAfYLqJYjof09CLiW3I9L5eoWceifD+OWlYCJGJwKd7/wCIAIAPaj6SfKTgG+g/wC6h7eLskoHf8jChaAPQPbbdz/lIY5CcoC6iAsiw2Sgwj4856PwHzqUcD9VACB/grDSsbgzCNER4niLYJnsJvMU2XQgXtrSUyX2Roi4FHA98FHgAsNDCGcVAmmJqKC8MAgAExYiMA78nCWMpIgzUSJNlIghJ2gUjuIxAoLLcKElhLiItzVCPUQkRJ0oURYjPjCGKPzZI4pzQypKwokNZEQUAbeJ30DLEFsMBSKw+KJHhKN+voGiImi2ii8SIoxSXITlEMGXQYjviJGw4SRwGIk57uVYlyBPOKKHiOghMBqnS0QdFZFIOhbijowgjxYOCYqmVC9nK0LIJUEWjUwqZv8Sm9TZ9EoWIqbkjDWtw4UPI6EE46GSkqMhW2o8yYzDsYaW2nolLrvxmFcKkBcg2pm6NPHKAPySFcVkYVCM88p4USOZieGFd5TpM3M0JprXQGM0qmmImG0Sm5Ug4yWu+Q1tRoOQfmMbIjj4CLHlgpzXQCdRzPmNVApikQSxkdNsIU5LCERJX+PHVHQAAB8EdBjc9Bt8DiTPTJRCbDYxKDsDEokbKMQcA1UAEA46iR0UtB5lsYAhLMqMkfXvj5cQaSFI2gy1rDIW/SyESOXjUYdAogWQdEzJDhkEAOBlorjIqSGAqggZ5M+dPgUAUSvhj5dyw6j06elPL+EVG4B0EVX/bcY3S/iJrMKiXtOiQHSOCYsGUGJ2TAOAVQ0B1kS4QKxqfZoyzeGABxpQECV4GlmFscuuZaI0qxDbWjuwV020FRFVaeg3WIeWvK5wEicASF3BKYzDRmeuvjBrLCzbCgiwlB+TpawvOCtTACgWItRB6j5CK1pekNZNy9DsVw/RSoSwtrW4IK1V15agIRQsF7IVhn7E9tt23Ba3wwgABthXjOCa4rXKfd9Ajovc2TLitBZRrT6oW93nfsO5oHhtQrjb3U+I1z1LKc1Sr0He8nbivIaYaWZtAV+CtNe9m6jvm2CpEv6y4774zYR+9TcMAGdiAiZZbYCz4gjGLiVU7oDnGYKtKZMJW1heqYCGuSAMCleMQqjxSMUqAgEAOw==)
.
Теперь изменим порядок суммирования и воспользуемся тем, что если
![](data:image/gif;base64,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)
, то
![](data:image/gif;base64,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)
далее следует
![](data:image/gif;base64,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)
.
В последней сумме коэффициент при
![](data:image/gif;base64,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)
равен 0, кроме случаев
![](data:image/gif;base64,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)
или
![](data:image/gif;base64,R0lGODlhKAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAjAAwAhQAAAAAAAB0AABwcHAAAMx0dNAAcSAAdSB0dSAAzWgAzWx1GbDMAADMAHTIAHTIAMjQdNDMeRzNGbjNbgEgdHUgdAEgcAEceM1ozAFszAFszM0hbSEhZf1l/WVtuSEhuf11/f2xGHWxZNX9ZM39/XX9uSHd3XXtqRG5/WXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZZQIBwSCwaj0hjIMlsFhfOKJIirVqvQ0zgAEhgASXBUjgGMBRCwTETaLsD6OID0CBavkivkAogEPVfGANDCEKAakhvbkYMUEJQGEMFeIRDDkNiXAyUIUNhXEEAOw==)
. Эта сумма сводится к единственному члену
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhQAAAAAAAB4AHh0AABwcHB0AHQAAMx0AMh0AMwAdSB0dSAAcSAAzWgAzWx1GbDMAADIAHTMAHTMAMzIAMjNGbjNbgEgdAEgdHUgcAFozAFszAFszM0hbSFl/WV1/f113d0huf0Rqe2xGHX9ZM3d3XX9/XX9uSHtqRHd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZ+QIBwSCwaj8jhI2CUJJ1JAASgKhqiAEEyc5xghYfj6FrkfoWMoiUQIA+nxAETsCaGy0e3EQRwKI0PRiJ6aEMXRA9+dnllBBUAJkWBRCILRiqEiUOKQpNEnERwQwpvABRDd0MFSBqcInEJQyINlFghZ0KvAAhYWlEPj0JtuLhBADs=)
.
Теорема. Число всех нормированных неприводимых полиномов степени n над
![](data:image/gif;base64,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)
задаётся формулой
![](data:image/gif;base64,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)
.
Доказательство: Возьмём
![](data:image/gif;base64,R0lGODlhRQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABCABAAhQAAAAAAAAAAHR4AHh0AAB4eABwcHB0AHR0AMgAAMwAdMgAcSAAdSB0dSAAzWgAzWx1GbDMAADIAHTMAHTMAMzIAMjMzWzNZfzNGbjNbgEgdHUgdAEgcAFozAFszAEgzM1szM0ZGRkhbSFl/WV1/f113d0huf0Rqe1luf2xGHW5GRn9ZSH9uSHd3XX9/XXtqRHd3ZmZ3d2aIiIBbM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwbKQIBwSCwaj8ikcslsOp9HBCBiDFivgQWRouRCvx+AxSgoQoaJJWvwfWoAM2NKe6w4pe388GzsPFN6TVRFEkMNRIVEBAFCG4xEeHmLAB0BaUozWEWXRyYAfEIRGUVYV0kSBaMcTA57QymcQ61Cb0SDegpDHKBHtyubRR0GoyxGEbxfo0IHS2VHM7FToMi3Qh6lVg91Q35Ls0eJhkOFGIFExUId0UWLAQxIHqCAQwTvRNp6puZDJ012erX2EWGzpFqbQwKheMlz7RGTIAA7)
,
![](data:image/gif;base64,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)
, подставим в предидущую формулу.
Теперь можно перейти к тестам неприводимости полиномов в
![](data:image/gif;base64,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)
.
Тест1. Полином
![](data:image/gif;base64,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)
степени
n>1 неприводим в
![](data:image/gif;base64,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)
тогда и только тогда когда
![](data:image/gif;base64,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)
для
![](data:image/gif;base64,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)
.
Причём если полином приводим то тест сработает достаточно быстро. Для неприводимых полиномов этот тест становится медлительным из-за вычислений НОД в
![](data:image/gif;base64,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)
. Для исправления этого создан
Тест2. Полином
![](data:image/gif;base64,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)
степени
n>1 неприводим в
![](data:image/gif;base64,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)
тогда и только тогда когда
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
для всех
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,R0lGODlhEAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAMABAAhQAAAAAAAB0AAB0AHQAAHR0AMgAAMx0dSAAcSAAzWh1GbDMAADIAHTMzWzIyWjNbgEgcAFozAFszAFszM0YzRkhIW0huf11/f2xGHX9/XX9uSHd3XXtqRHd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwYzQIBwSCQuAsUkI1k8MIeYJ/EBEASQxYIUEOVII1fDc6kZWJgIIeSpGAYI58h2LpzQ59ggADs=)
- простые делители
n.
Алгоритм Берлекампа разложения на множители над конечными полями. Идея Берлекампа основана на китайской теореме об остатках для полиномов:
Пусть
![](data:image/gif;base64,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)
- полиномы из
![](data:image/gif;base64,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)
, причём
![](data:image/gif;base64,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)
взаимно прост с
![](data:image/gif;base64,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)
при
![](data:image/gif;base64,R0lGODlhJwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAiAA8AhQAAAAAAAB0AAB0AHQAAHQAAMx0AMh0AMx0dSAAcSAAdSAAzWgAzWx1GbB1IbjMAADIAHTMAMzMzWzNbgEgdHUgcAFszAFozHVozAEhIW11/f0huf0Rqe2xGHX9ZM39/XX9uSHd3XXtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZdQIBwSCwaj8ik0vgILAGWwnNKpSKqScqx8wwksMPKBCAIOI+NYsfMNjOKhudFiUFyAXEkaHlFmqVgbWxFEEIDSRVJHkkmYUkHSRFJFg5DAQRoYE+FAJCaS00Kn0tBADs=)
. Пусть
![](data:image/gif;base64,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)
- произвольные полиномы из
![](data:image/gif;base64,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)
. Тогда существует единственный полином
![](data:image/gif;base64,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)
, такой что
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
. Это же можно сформулировать на языке отображений:
Отображение, ставящее в соответствие полиному
![](data:image/gif;base64,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)
вектор
![](data:image/gif;base64,R0lGODlhaQAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABABkABAAhQAAAAAAAB4AHgAAHR0AAAAAMx0AMx0AMgAdSAAcSB0dSAAzWgAzWx1GbB1IbjMAADIAHTMAHTMAMzIAMjQdNDQ0HTVIWzNZfzNbgEgcAEgdAEceM1ozAFszAFszM0ZGM1tINVtISFlGRl1/f113d0Rqe0huf1luf25IHWxGHX9ZM25GM39ZSHd3XX9/XX9uSGZ3d3d3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikcslsKjOBgHMomVqvyaoTFR1mFtcCU4AtH8VLcjEhzFy15rhz0nwVU16wkzPFy7F8Til6QmwAbkUPUQ1HJkUBAyUAKgEIQwd/RopSRxCORJCSM5WXQ35ChKhJaEYLp0MPGJmmQ7JGrER4KQyJRIaIQ3ZIKbhEBEcPs0QGScRIx5q0bamFSMVeyIzK1dZIwEQP2gC/R4EArDPXQhXI2yJD4kTpSOzRQqdf2EK8RhBJCkYwKUPEgV+/Iw4AAFxBRCC3b5tYbSgijotBTkP4EWBlSUhHAB87boQVBdcAAAYrBrhYxCA+akUmEongBM62IxZm1lzTZonMKWACJClJBsvglAdGj1Ao8iLoEqLShET52MSmEas3qyqxOtUJxqxgMwUBADs=)
, где
![](data:image/gif;base64,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)
, является биекцией между
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
.
Доказательство: Проводится расширенным алгоритмом Евклида. То есть определяются полиномы
![](data:image/gif;base64,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)
, такие что
![](data:image/gif;base64,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)
. Полагаем
![](data:image/gif;base64,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)
. Тогда
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,R0lGODlhNwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAyAAwAhQAAAAAAAB0AAAAAHQAAMwAcSB0dSAAdSAAzMx00NAAzWx0zWh0zWwAzWh1GbDMAADMAHTIAMjNbgEgcAEgdHUceM1ozAFszAF1dRl1/f0huf2xGHW5bNX9/XX9uSHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZ7QIBwSCwaj8gksqBsOp+AR4DZ3Bw3BKcCSk0GoMVNYPvcdIuWgAM8TLOF5+FXKZUUJwE70iPIivJDcQABgEoVRoRQCBBEgnJJHkkXakkJRmZKYmtGm15HFgdHjkZkQxRlUBajTREYb0INRlIBA0RihLhjRXO7uYlEF0ZBADs=)
. Если бы нашёля такой
![](data:image/gif;base64,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)
, который бы был решением этих сравнений, то полином
![](data:image/gif;base64,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)
должен делиться на все
![](data:image/gif;base64,R0lGODlhJwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAiABAAhQAAAAAAAB4AHh0AAAAAHR4eAB0AHQAAMx0AMgAdMwAdSB0dSAAzWgAzWx1GbB1IbjIAHTMAADMAHTMAMzIAMjQ0HTIyWjNZfzNbgEgcAEgdAEgdHVozAFszAFszM0gzSF1/f113d0Rqe0huf1luf2xGHW5IHX9ZM25GM3d3XX9/XX9uSHtqRGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwaHQIBwSCwaj8gkcpJkKp+Hp+CZpFABiGuRowUwiIPABRCIGiGj4StAGAaGqywRUhBmjubit1GMEF9FDkUleXADgn2IRgZ4R3dGfkNWQlxGL4VCKF6QGEMLlJhDEF1yQgGnW4FuQxoBD0MlRBtaEldOACxJHkUrAiJKEZ0AHacKXbdGyE9vXUZBADs=)
. Поэтому
![](data:image/gif;base64,R0lGODlhrwAkAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABACpAB0AhQAAAAAAAB0AAB4AHgAAHR0AHR4eAAAAMx0AMgAdMwAdSAAcSB0dSAAzWgAzWx1GbDMAADIAHTMAHTMAMzIAMjQ0HTMzWzNdXTNGRjNGbjNZfzNbgEgdAEgcAEgdHUceM1ozAFszAFszM0gzSF1/f113d0Rqe0huf0Rqalluf1V7am5IHWxGHX9ZM25GM3d3XX9/XXtqRH9uSHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikcslsOp/QqHRKrVqvgFVgy+16udiwdCK+krGC5HajDjzMyfPUdmAOyoC68l7UX7dJEAFKAQRXfkh8QopDMVcIVxlPMkUUZWlIgmxIhVaWTJBJBUhvTg1VowAgpUssp6p4mEeChIZUIE+vACxDIVybQxFFNp1Cg0IyoVJgwkTEtgDHAMonQ4YcAafYuh2ARCBdgQHARoXVRgIBGtGIROdDhSYAxAoAlETKuEftQ6cORRCK+PryDwkdJP6IBMwzJJ2tTjZsdRjCYYEQEPUA+BJXK0kEA0ImGuEnpAEvhSiHvCNCUoYAVkMgwHzSjOGRlwB3IZIla9TB/4b2UgmRZUSTGmjDisxkQRIAUSEyYw6xaIQpEpE5iQz0UnAIC6pWj2CN+WYn0KFCEEU1m0nakWJG9AkRyvJqUYVvsHJARNIFAF0xyT1h1fdvzp9oEzudp05qRHmKi447unLIJ1hGEBepUBTYQkgfhIQOBkWZkK1dugoBLaTmEBtCGBQJ5Zrn2b1CWGSE4OepQrflkB4B0dR1Edn4kkcjUPl0EWnYWJ2M8qzXc4puhnQtJSgAhnTeHVbDBrz7r1mTOQmH522IXCGstHSVNl2I6iGjh0hwcvnKfib9/YcHeh0Z4YETAiohBwALLsFIJoJZ8eBdRMgwAGQD/hYhPMwV4XIIEiIUYSGGs3iWRH5ENFiEiliwKISLLpZBy1ECbZFREzEuGKMvN2bo44/obWjMeoQAaeSRSKZDCHBINulkkwQ958VMT1Zp5ZVYZqnlllx26eWXYIYp5phklmnmmXi8h+aabLbpJh4hvinnnHTWWQ6YQQAAOw==)
.
Теорема. В поле GF(p) – поле Галуа (конечное поле, содержащее p (простое число) элементов) имеет место разложение:
![](data:image/gif;base64,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)
.
Доказательство: В поле Галуа
![](data:image/gif;base64,R0lGODlhLAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAmAA0AhQAAAAAAAB0AAAAAHR0AMwAAMwAdSAAcSAAzWgAzWx1GbDMAADMAMzMeRzMzWzIyWjNGRjNZfzNbgEgcAFozAFszAFtISF1/f0huf113d2xGHX9ZM39/XW5/WXd3XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZYQIBwSCwaj8jkcAEQKJ/QoQHgiFqPDcDmyh2KumAhJTAIKxeBwPBhJqIDiiHCXUZW0vh0wjj/DgltXQeBSINCFkNxUHd5ekcURpCEQxNiewBvBZNLaZoAQQA7)
(а также по малой теореме Ферма) . Значит s является корнем полинома
![](data:image/gif;base64,R0lGODlhLQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAApAA0AhQAAAAAAAB0AAAAAHQAdSB0dSAAzWgAzWx1GbB1IbjMAADMAMzMeRzQ0HTMzWzIyWjNGRjNZfzNbgEgcAEgdAFozAFszAF1/f0huf0Rqe2xGHW5IHX9ZM25GM39/XW5/WXd3XWp7VX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkUQERIJTQKLEirUoZwqd1W4wAFtywsBIYiJWl8jBwLpYzgDShaAAchApzW2jQHEVOVgGDhGxGAkkTe0iKRx0AdYtFDZJcBW5DFAEJixsBd0KGAEEAOw==)
, то есть (
x-s) является делителем
![](data:image/gif;base64,R0lGODlhLQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAApAA0AhQAAAAAAAB0AAAAAHQAdSB0dSAAzWgAzWx1GbB1IbjMAADMAMzMeRzQ0HTMzWzIyWjNGRjNZfzNbgEgcAEgdAFozAFszAF1/f0huf0Rqe2xGHW5IHX9ZM25GM39/XW5/WXd3XWp7VX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj8jkUQERIJTQKLEirUoZwqd1W4wAFtywsBIYiJWl8jBwLpYzgDShaAAchApzW2jQHEVOVgGDhGxGAkkTe0iKRx0AdYtFDZJcBW5DFAEJixsBd0KGAEEAOw==)
. А так как это выполнено для всех
![](data:image/gif;base64,R0lGODlhSAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgBDAA8AhQAAAAAAAAAAHR4AHh0AAB4eAB0AHQAAMx0AMgAdMgAcSAAdSB0dSAAzWgAzWx1GbDMAADIAHTMAHTMAMzIAMjMeRzNZfzNbgEgcAEgdAEgdHVozAFszAEYzRltISF1/f113d0huf0Rqe1luf2xGHX9ZM25GM25GRn9ZSHd3XX9/XX9uSHtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwboQIBwSCwaj8ikcslsDjeBaINIilqjCuKEuXU6S4HFEBZ4jAWhIWl8cA68zYBxTcweKXCEl2BtE1cEDkdpQhtDJ0VmcFNDEFGKQ2JJG35KEUiXRnwWAAGVQnpDUzBFHUsYRRl9Q1eMADCfRBEFFwCoRBBEoUWOV7EQrqyQG4x0Q7FDCUMYkAAQkKRGFUu5Rp8QtUKEACTIQtkABrxFhkUaSxwCRRjB6kjeeIWCuEK3APNs1J8E4OVJ20YoGQnVy1unVUY4XJEEgE8YJByOWMGnhiKci0skmMPIsSORN02weRyJZItCOUyCAAA7)
то
![](data:image/gif;base64,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)
. Также следует заметить, что
![](data:image/gif;base64,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)
и это два нормированных полинома, из этого всего и следует их равенство.