BEGIN
![](data:image/gif;base64,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)
END
END
2. FOR i=0 TO m-n // выдача результатов
BEGIN
RETURN
![](data:image/gif;base64,R0lGODlhEAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMACQALAAsAhAAAAAAAAB0AAAAAHQAAMx0dSAAcSB00NAAzWh1GbDMAADIAMjMzWzNbgEgcAEgdHVozAFszM0YzRl1/f0huf2xGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwUtICAqQQI0YiqkRCo6aeaK7TsDSDqkEgClhxtAEMiZRIbZQ8h0kQIpRjNTAoQAADs=)
![](data:image/gif;base64,R0lGODlhLAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMACQAoAAsAhAAAAAAAAB0AAAAAMx0dSAAcSAAyMh00NAAzWh0zWh1GbDIAHTMAADIAMjMzWzNbgEgcAFozAFszM0YzRl1/f0huf2xGHW5bNX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwVKICAMABCU6BYoaOu6UHC+AAEYC41Gci8jLoGuZAEkhsgXIxkpJJ8oErIBLfl6LmB1q7NILc7XY+dSsaqxGZnLhkrabrhcp57TZiEAOw==)
RETURN
![](data:image/gif;base64,R0lGODlhDQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIACQAJAAsAhAAAAAAAAB0AAAAAMx0dSAAcSAAyMh00NB0zWgAzWh1GbDIAHTIAMjMzWzNbgFozAFszM0YzRl1/f0huf2xGHW5bNX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwUhICAMABCUKAEYCwpQAOKWTzGXzBzdszPbvKCwwcsEFIAQADs=)
END
Очевидно что доминирует первый цикл, который выполняется m-n+1 раз. В каждом цикле происходит одно деление и пересчитывается ряд коэффициентов. Таким образом трудоёмкость алгоритма PDF есть O[n(m-n+1)]. Это как раз то время, которое нужно для вычисления произведения
![](data:image/gif;base64,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)
над полем.
Наибольшие общие делители полиномов над полем. Дадим следующее
Определение. Пусть К – область целостности и
![](data:image/gif;base64,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)
, причём
![](data:image/gif;base64,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)
.
Полином
![](data:image/gif;base64,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)
называется Наибольшим Общим Делителем (НОД) полиномов
![](data:image/gif;base64,R0lGODlhKAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQAjAA8AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHUceM1ozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgV/ICCOZGmeaJoaKjC0cMvFZ8XE2UIKQQQErFOjFCBcAJ5AYnQoNQoiyckTJGVEOpKD5CndTNXdaYt6mTLhkdTk+AIeo4o6nBZBxiSFqBKmkIZeAHoaJE0jAYgtciIZAVk/JY8TKX5/MXAiMzQiZilkABWISyeVJZgnpzCho5skIQA7)
и
![](data:image/gif;base64,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)
если выполнены следующие условия:
1.
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
2. Если
![](data:image/gif;base64,R0lGODlhKQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAlAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSB00NAAzWh1GbDMAADIAHTIAMjQ0HTNGbjNbgEgcAEgdHVozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwWGIBCMJGCeKOqk6cqi0ItyhYwKdn66OmDYsdyid1IQGYEFJQJopDiBwSlwOP0AFNJoaCKcajSZ0Yhi6CSv2ovALbdhgPCJeVGz0CxzjgyQmuwpD3wnDEwsMRQnCSkWTwAaJ40mV1ojJwQBRm2JKJkmEkknF0QmEylOpKk8Jzg6ej1IAS+rOiEAOw==)
,такой что
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
,то
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,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)
.
Отсюда виден так называемый алгоритм Евклида для нахождения НОД двух полиномов, также использующий теорему делимости, который работает следующим образом:
![](data:image/gif;base64,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)
, при этом
![](data:image/gif;base64,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)
. . .
. . .
![](data:image/gif;base64,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)
, при этом
![](data:image/gif;base64,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)
Так как
![](data:image/gif;base64,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)
, то очевидно что эта последовательность закончится самое большее за
![](data:image/gif;base64,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)
шагов. При этом справедлива следующая
Теорема. Последний отличный от нуля остаток
![](data:image/gif;base64,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)
это и есть НОД(
![](data:image/gif;base64,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)
).
Cледует учесть что НОД может быть определён не однозначно если в области целостности имеются обратимые элементы.
Теперь пусть имеется некоторое поле F,
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,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)
. Применяя PDF можно вычислить НОД(
![](data:image/gif;base64,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)
).
Пусть
![](data:image/gif;base64,R0lGODlhIQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAcAA4AhQAAAAAAAB0AHQAAHR0AABwcHAAAMx0AMgAcSB0dSAAzWh4zRx1GbDIAHTMAADIAMjQ0HTMzWzNbgEgcAEceM1ozAFozHVl/WV1/f0huf2xGHW5GM39/XX9uSG5uRGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZ1QIBwSBQ+isUjcgkwMIcdwZOonAoP1oqVqCEGFMIuoIEMDISgAGKIBYAZw0xz2RUTHRIh+QrQOJcESw5oQ1pCf0t5RQ5wQyAFQ4hFEGCLegAdABQAEWNLcAkAG4JqAAR/FY19X0ITAUWVU3JbVVNSVoNburVBADs=)
и
![](data:image/gif;base64,R0lGODlhIQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAbAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh0zWh1GbDIAHTMAADIAMjMeRzQ0HTMzWzNbgEgcAFozAF1/f0huf2xGHW5GM39/XX9uSG5/WWaIiIBbM4iIZgECAwViICCOpOiU5ImuQMGOwpvKo0FTNJkAXTCMlB0D5RP1DiKbqLEYRVqrC0BKagBLF9eKgGKOkJqRdiUpWb+lcQmyq5IC6iGqiQBgaiQkidIUXQJtEwEjF20yFjmJADEyDWWKiiEAOw==)
- некоторые произвольные полиномы из
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAfAA4AhAAAAAAAAAAAHR0AAAAcSB0dSAAzWh1GbDIAHTMAADQ0HTNbgEgcAFozAEhuf11/f2xGHW5GM25GRn9ZSH9/XX9uSG6AbmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwVpICACQWkGBMCU6eiK7AsI76HKuNiO0O4yhtyrJ7PJGEKZD4AYFX7BUYAG4/kwpxdSFoyOll6jCCgbiEXEUWI00eK2Lx/17RVFAHVAWpT/vTBCLQMxdFIjK2J7STeLX41jfTiKSSsBizEhADs=)
. Тогда справедлива
Теорема. Если
![](data:image/gif;base64,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)
НОД(
![](data:image/gif;base64,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)
), то в
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAfAA4AhAAAAAAAAAAAHR0AAAAcSB0dSAAzWh1GbDIAHTMAADQ0HTNbgEgcAFozAEhuf11/f2xGHW5GM25GRn9ZSH9/XX9uSG6AbmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwVpICACQWkGBMCU6eiK7AsI76HKuNiO0O4yhtyrJ7PJGEKZD4AYFX7BUYAG4/kwpxdSFoyOll6jCCgbiEXEUWI00eK2Lx/17RVFAHVAWpT/vTBCLQMxdFIjK2J7STeLX41jfTiKSSsBizEhADs=)
найдутся полиномы
![](data:image/gif;base64,R0lGODlhIQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAcAA4AhQAAAAAAAB0AHQAAHR0AABwcHAAAMx0AMgAcSB0dSAAzWh4zRx1GbDIAHTMAADIAMjQ0HTMzWzNbgEgcAEceM1ozAFozHVl/WV1/f0huf2xGHW5GM39/XX9uSG5uRGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZ1QIBwSBQ+isUjcgkwMIcdwZOonAoP1oqVqCEGFMIuoIEMDISgAGKIBYAZw0xz2RUTHRIh+QrQOJcESw5oQ1pCf0t5RQ5wQyAFQ4hFEGCLegAdABQAEWNLcAkAG4JqAAR/FY19X0ITAUWVU3JbVVNSVoNburVBADs=)
и
![](data:image/gif;base64,R0lGODlhIQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAbAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSAAzWh0zWh1GbDIAHTMAADIAMjMeRzQ0HTMzWzNbgEgcAFozAF1/f0huf2xGHW5GM39/XX9uSG5/WWaIiIBbM4iIZgECAwViICCOpOiU5ImuQMGOwpvKo0FTNJkAXTCMlB0D5RP1DiKbqLEYRVqrC0BKagBLF9eKgGKOkJqRdiUpWb+lcQmyq5IC6iGqiQBgaiQkidIUXQJtEwEjF20yFjmJADEyDWWKiiEAOw==)
, такие что
![](data:image/gif;base64,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)
Доказательство: Из всех полиномов вида
![](data:image/gif;base64,R0lGODlhkwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQCPAA8AhQAAAAAAAB0AAB0AHQAAHR0dABwcHAAAMx0AMgAdMgAcSB0dSAAzWh0zWh4zRx1GbDIAHTMAADIAMjMeRzQ0HTMzWzNbgEgcAEgdHUceM1ozAFozHVtISFl/WV1/f0huf1luf2xGHW5GM25GRn9/XX9uSG5/WW5uRGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwSJcakcslsOp9DpFIKBVCr1YNzkG1ysWCilvl9jslh4XVZSivbzIy7uFYinvUkHKyphhhNGg9Pf3NPfVCATIiEikICARYAAWdEEB9FAQSYKQEKQ3dCAUMhQxCZmwCdn0KhSXJvkAAalEqXqJyeACWhnQRDiLdEmrmsAK5CEAVCF0mVpEKOQhHRAINEKc8ApYVE1EqwS8rM10Xa29VD39PlFQDn3NIA66pF5UIh5wACSeunrUPO8TNCb1aAgwgDyBOSYEgzI/mUDPRWjlGKbfomeltSRoySh0UiXGOlQVFEIyAppmEEoKPHJCmnFTEmRF9MdlGABRySTQmFwn5DBl2UVDPJT4JLwjlRFGGUkJ5GCRKth03f0ZBEFgjRcEbpP3sAtIogEuqfAznuAHwNGhbAWFBJmWgtesRUkkFi4Q6rtJZtXr2iDhZRao2UwiFOhzAddULAGUYhDosqsnAIYSKRD1YmUi6yo8SlsN01/JnyEAxQ+i7JY0R1EtZOUKfG88T16yF7jBAuMQATk4Jofs+Zm+QiEd6+lUSYygR5k3W0dOm+vRoMbNiBDtIkUvm6dSbYMxkaT15J5kjlyQcBADs=)
выберем любой из полиномов наименьшей степени и обозначим его
![](data:image/gif;base64,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)
. Если
![](data:image/gif;base64,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)
не делит
![](data:image/gif;base64,R0lGODlhKAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQAjAA8AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHUceM1ozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgV9ICCOZGmeaJoaKjC0cMvFZ8XE2UIKQQQErFOj5AkQRoHRodQoiCQnT9Ck040cJE/pZpqWBNxr6mXKeEnQkiP8GFVGEu9ZpAFYxSOFqOKlkIYoeiRLSAFJKm8khwASAWF3Eyl+fzFtIjM0ImQpWHuGCSiTJZYnpDAVAaCZJSEAOw==)
, то
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,R0lGODlhOQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwA0AA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSB00NAAzWh1GbDMAADIAHTIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwWwICCOpOiU5ImurBgEbVnEgEDH2wwwt0obN1aEdIhNgiJFsFKyiTICpogBOwIaqMBAtAkUAUDaopQj8USEsUWnK1kAb/PqRX/JzBBARJmEt0sEKAxjLFYof4EAbUMifyV5ciwZLU4AUBWGFluNLA98I2ckFnV0n3pEfRsACCNYKGOsFyNhK6ZkoCMEMHEiE4RwAXwRMCIWtiM+LKPEKFIjzkjRPSiVLAyQ0jRUwdndNCEAOw==)
,
![](data:image/gif;base64,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)
. Но тогда полином
![](data:image/gif;base64,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)
имеет вид
![](data:image/gif;base64,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)
, в противоречие с выбором
![](data:image/gif;base64,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)
.
Из теоремы следует, что для взаимной простоты полиномов
![](data:image/gif;base64,R0lGODlhKAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQAjAA8AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHUceM1ozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgV9ICCOZGmeaJoaKjC0cMvFZ8XE2UIKQQQErFOj5AkQRoHRodQoiCQnT9Ck040cJE/pZpqWBNxr6mXKeEnQkiP8GFVGEu9ZpAFYxSOFqOKlkIYoeiRLSAFJKm8khwASAWF3Eyl+fzFtIjM0ImQpWHuGCSiTJZYnpDAVAaCZJSEAOw==)
и
![](data:image/gif;base64,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)
необходимо и достаточно чтобы
![](data:image/gif;base64,R0lGODlhrAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQCnAA8AhQAAAAAAAB0AHR0AAAAAHR0dABwcHAAAMx0AMgAdMgAcSB0dSAAzWh0zWh4zRx1GbDMAADIAHTIAMjMAMzMeRzQ0HTMzWzNbgEgcAEgdHUceM1ozAFozHVtISFl/WV1/f113d0huf1luf2xGHW5GM25GRn9/XX9uSG5/WW5uRGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikkihZOp/FJlIKBVCN1yoRozUeloLqF9wtE8fJMBSNVJshAYUZeponEXZtFol37o99ZiNzGw9PIwxLhVWJSBp5QxuMipNzckQDARcAAWxEEUUBBCEALHFDgaaNAIagRKYEQwFDdXOZm51KpEKwsrRDvUKSQrulor5CtYJHEQVCXEYsngCDAKtCELzUQxyl00KJ1wDZjkvNz0nf4UXk2IZCFgDq1uxHAff4s0aXwEXvRd8GHGlHDCCSAf+GQEjIKd+9Iyz8HRkRkOG4f8NKUatoZOEyJW6IUDwCjd2/SxsSjSQ5MMkjKCENGilJhCAAfkK+0VRoB6eV10hDMLD5VqGlsCEGhnwjQY/dpjJUMsL5J80IU3Hjnmp8Ne1qR63CHN4TV43IAmFsXgpxRcTQWaaohrhy8CmakLNFAhlRewTvBk9X2BbBS0QvJzaCtxKOa2aDT3yUhIwIsErfkFVw5AwYk1GIZQwB/pUtA9lIt7bIAIAWXSSikX+fQw8ZrQXOMSEZXBZJPOUJ795mclf5jcX3kz9mlMFcYhOkEwhgqyg/Ev2EgGIDoyOxjv0rpLCnjvAdgvynHiXlFd3zeRn4kfRdJnyHsiH8/PvfS+PfryUIADs=)
для некоторых
![](data:image/gif;base64,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)
.
Неприводимые сомножители полиномов. Для начала нужно сформулировать ряд известных теорем:
1. Основная теорема алгебры. Каждый полином
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
из
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhQAAAAAAAAAAHR0AAAAAMwAdSAAcSB0dSAAzWh1GbDMAADMAHTMeRzQ0HTMzWzVbbjNbgEgcAFozAEYzRkhuf0Rubl1/f113d2xGHX9ZM25GM39/XX9uSHd3XXd3ZmZ3d26AbmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZ0QIBwSCwaj0hAICAYRpaFpHBpIGYCCKegAshKidVhwMgARLxfACbMGaCN728Y0ERGjEzxcC28S+NCWXFhCoBOhgMJRXwACl+GfkVzBEmRRBpdi3MDlEhxIklzZksBnX1vY30BikKMaYZIopVpe7JJUF9UAEEAOw==)
- поля комплексных чисел
![](data:image/gif;base64,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)
имеет корень в
![](data:image/gif;base64,R0lGODlhEAATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAMAAwAhAAAAAAAAAAAHR0AAAAAMwAcSAAzWjMAADIAHTMeRzMzWzNbgEgcAFozAEYzRkhuf11/f113d2xGHX9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwUrICCOZCk2QSA8wEIGpfCa5EArorTSIlPwogMwaBg2CCZHDTlCmBipAFMUAgA7)
.
2. Отличный от константы полином
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
из R[x] неприводим если и только если он имеет степень 1 либо это квадратный трёхчлен с отрицательным дискриминантом.
Имеет место обратное утверждение.
Теперь для полиномов над полем K – поле.
3.Если неприводимый полином
![](data:image/gif;base64,R0lGODlhVAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgBQAA8AhQAAAAAAAB0AHR0AAAAAHQAdPQAAMx0AMgAdMgAcSB0dSAAyMgAzWgA0SB1GbD0dADMAADIAHTIAMjMeRzQ0HTMzSDNbgEgcAEgdHVozAEYzRkZGM0hbSFtuSF1/f0huf113d2xGHW5GM39/XX9uSG5/WXtqRGZ3d3d3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwaj8ikcslsFiVKqDMZqA6rhUc1sdxOAYYlSfBFBo6XaegrZR7KxRAXkJkDLoypfZlRH0kDVQFhSQ4AEEVpTmtCEGcAA4SRQhFGAQRCKgF2b0Qqe0pkhER4RZeZm0N7HEMdRaNEa4xEEBZwAJqYRopxALNCvwhDC6q+sEQDR4i0glXHRA0Xz6VGyUZzGQQfdIR9Qs9DtkUQhnDLAKC8RuJDs7znUirgABR540UYTqMQe9RF9deEKCgmJM+2SkYMDRSxZF4RewBMDLCjbohCAAyBaRTijU6AchnK+Qpg78KjIRCJNHsmaE4zIf5CkIR5UgioJduYtLmVpKKSFl9NyCw5x7NnyiU3iwLYqRSNlZ9PgwAAOw==)
делит произведение
![](data:image/gif;base64,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)
то
![](data:image/gif;base64,R0lGODlhUwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQBNAA8AhQAAAAAAAB0AHR0AAAAAHR0dAAAAMx0AMgAdMgAcSB0dSAAyMgAzWh1GbDMAADIAHTIAMjMeRzQ0HTMzWzMzSDNbgEgcAEgdHUceM1ozAEhbSFtuSF1/f0huf1luf2xGHW5GM39/XX9uSG5/WWZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwb/QIBwSCwajZCjcKJsOp9QogEqiFqJpqmzeu0ejsyueJxpXBnjpxn6GToCFcBAOxc+iqYAYRgYisBHA3AAAVpGd0R6HQB5CUNfdgAKAEmTSVlKaGhEDlcPBXEWSoZEDG1FDnEmABSMABGuQqREA2uctktHq0S4Q7NCA0edABl7xFoZQh+/QqJGw1DJQ1xFy0fOqGbY0EmyRyAAm5xxRGFF3cRDFqTMEsJ4jodGu5NFkFcZpBhEiEb1RJBskSM0RBofdQFsnSpiLlGAPk4MfgggDuIQcVD6PUH35EKTffyicIxCrUkqQET+dCn5bszIc1YyPIxnBCRMJS/TdGn4RCbNAzFBAAA7)
или
![](data:image/gif;base64,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)
.
4. Пусть
![](data:image/gif;base64,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)
. Тогда полином
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhAAAAAAAAB0AAB0AHQAAHR0dAAAAMx0AMgAdMgAcSB0dSAAzWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf1luf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwV0ICCOZPmU5ImuY2ew42DBqEoDx00yOoCVggCAEniJGqgAYSQU5UqFEc/FWgCsI0eEhBhJRkaUgEdykCijjcwXLn1LDjLARsEC2qPMFU5SgG8dKxskGAGGdiJoJE0AEgFyiDMsSDo2AH40AzdmTAEJN5YmJCEAOw==)
может быть однозначно представлен в произведение неприводимых нормированных полиномов над K[x]. Разложение является единственным с точностью до порядка сомножителей.
Назовём полином
![](data:image/gif;base64,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)
примитивным, ecли его коэффициенты – целые числа, НОД которых равен 1. Тогда любой полином из
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhAAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAF1dbll/bkhuf113d11/f2xGHW5GM39/XX9uSGZ3d3d3Zm6AbmaIiIBbM4iIZgECAwECAwECAwV8ICCOZGmeKBCsI7MWqfiSW4CMwXjEJYyXDRhjxxNNfIAEcdQgiJZFpODkZJgCU9noKLKaErrTDgrwJW6lHGB4GqC3vsEJSe7ORcpSdn2iAMhcABMBCz8kZBspSA0rAWRshms2cCQOKXWKIwkBTiZeRYI+ThYDjWkBSCgzIQA7)
ассоциирован с некоторым примитивным полиномом (два полинома называются ассоциированными, если один из них является скалярным кратным другого). Верна теорема
5. Произведение двух примитивных полиномов из
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhAAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAF1dbll/bkhuf113d11/f2xGHW5GM39/XX9uSGZ3d3d3Zm6AbmaIiIBbM4iIZgECAwECAwECAwV8ICCOZGmeKBCsI7MWqfiSW4CMwXjEJYyXDRhjxxNNfIAEcdQgiJZFpODkZJgCU9noKLKaErrTDgrwJW6lHGB4GqC3vsEJSe7ORcpSdn2iAMhcABMBCz8kZBspSA0rAWRshms2cCQOKXWKIwkBTiZeRYI+ThYDjWkBSCgzIQA7)
снова примитивный полином.
Доказательство: Пусть p – простое число. По определению примитивности для простого числа p имеем:
![](data:image/gif;base64,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)
,
![](data:image/gif;base64,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)
, откуда
![](data:image/gif;base64,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)
Иначе говоря никакое простое число не делит все коэффициенты многочлена
![](data:image/gif;base64,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)
что и доказывает его примитивность.
6. (Gauss) Если
![](data:image/gif;base64,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)
, причём
![](data:image/gif;base64,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)
, то
![](data:image/gif;base64,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)
, где
![](data:image/gif;base64,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)
и
![](data:image/gif;base64,R0lGODlhIwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABQAdAA8AhQAAAAAAAB0AAB0AHQAAHQAAMx0AMgAcSB0dSAAyMh00NAAzWh0zWh1GbDIAHTMAADIAMjQ0HTNbgEgcAEceM1ozAF1/f0huf2xGHW5bNW5GM39/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZ0QIBwSBxCisikElBYDgdO4jFKHVaqQsxQ0AQEipdigBD+BA5DwxABSDiIXeRCW3xIAFpGERMvCpIPQhVoSH1EE4ANAFNMQx+GQxGAUW9KbEVqAHdCFEWKWQELQ19EokWdRJVUjESoQxwDYU6BWItOrGK1SkEAOw==)
- полиномы, ассоциированные с
![](data:image/gif;base64,R0lGODlhIQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAcAA4AhAAAAAAAAB0AHQAAHR0AAAAAMx0AMgAcSB0dSB00NAAzWh1GbDMAADIAHTIAMjQ0HTNbgEgcAEgdHVozAF1/f0huf2xGHW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwV1ICCO5OiU5ImuwFawYibApUoDxi0uOmCVjABgMmqUAgPRJnAY5UQEnuXVopJ+PyAEEFGMvCIribBihEeR0ZS1BS6IIktyJB49wCQzwLsBIEoVJTx/FysEQlkjcD4BYBFCJHgjgSRGOjYmKzM0eiNBjSuYNSQhADs=)
и
![](data:image/gif;base64,R0lGODlhIAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAaAA4AhQAAAAAAAB0AHR0AAAAAHQAAMx0AMgAcSB0dSAAyMh00NAAzWh0zWh1GbDIAHTMAADIAMjQ0HTNdXTNbgEgcAFozAEhOM11/f0huf2xGHW5bNW5GM39/XX9uSGZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZpQIBwSIQQjwAjEllYHgUYp1ApHRqkleoxMxw0QQGAAxkgDAMHK1EjBjSRXO7xMREqhpk3crB8CLNEekh1cw1JR4JEEX1CC0djSIYIABtDHX9LhkIZAY4AFGFEnksOUVqHVQKnfqdaVABBADs=)
соответственно.
Полином в
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhAAAAAAAAAAAHR0AAB0AHQAAMwAcSB0dSAAzWh1GbDMAADMeRzQ0HTNGbjNbgEgcAFozAEhZf1l/bkhuf11/f2xGHW5GM39/XX9uSHtqRGZ3d2aIiIBbM4iIZgECAwECAwVrICCOZGmeJhSsbAE8q4GOMbmUwojMpowGo8eON6r4TIPSkCg6khTLlymQEwFFRtPjFNUBuk4AxFUSngaJUrYomHBP2x4pkNaeLF/1cbDMEEhdHDM+CiwsR2Y0QXRFYTNdPI5wTI2UADU8NSEAOw==)
неприводим если он не разлагается в произведение двух полиномов с целыми коэффициентами. В силу вышеприведённой теоремы видно, что полином неприводим в
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhAAAAAAAAAAAHR0AAB0AHQAAMwAdSAAcSB0dSAAzWh1GbDMAADMeRzQ0HTNGbjNbgEgcAFozAEhZf1l/bkhuf11/f2xGHW5GM39/XX9uSHtqRGZ3d2aIiIBbM4iIZgECAwVtICCOZGmeZhSsbAFAq4GO60EypTAms2mjgRGE1xtZfqZBiVgUIUkL5ssU0ImCoqMJcpLuAN4nIOIqDU8DRUlrFFC6J66PFFBvTxfwGjlgaggkXh0zPwssLDIiZzRCdUZiM149kHFNj5YAMUWaIQA7)
, если и только если он неприводим как полином из
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhAAAAAAAAAAAHR0AAAAAMwAcSB0dSAAzWh1GbDMAADQ0HTNbgEgcAFozAF1dbll/bkhuf113d11/f2xGHW5GM39/XX9uSGZ3d3d3Zm6AbmaIiIBbM4iIZgECAwECAwECAwV8ICCOZGmeKBCsI7MWqfiSW4CMwXjEJYyXDRhjxxNNfIAEcdQgiJZFpODkZJgCU9noKLKaErrTDgrwJW6lHGB4GqC3vsEJSe7ORcpSdn2iAMhcABMBCz8kZBspSA0rAWRshms2cCQOKXWKIwkBTiZeRYI+ThYDjWkBSCgzIQA7)
. При этом справедлива теорема
7. Если
![](data:image/gif;base64,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)
- полином в
![](data:image/gif;base64,R0lGODlhIwAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAfAA8AhAAAAAAAAAAAHR0AAB0AHQAAMwAcSB0dSAAzWh1GbDMAADMeRzQ0HTNGbjNbgEgcAFozAEhZf1l/bkhuf11/f2xGHW5GM39/XX9uSHtqRGZ3d2aIiIBbM4iIZgECAwECAwVrICCOZGmeJhSsbAE8q4GOMbmUwojMpowGo8eON6r4TIPSkCg6khTLlymQEwFFRtPjFNUBuk4AxFUSngaJUrYomHBP2x4pkNaeLF/1cbDMEEhdHDM+CiwsR2Y0QXRFYTNdPI5wTI2UADU8NSEAOw==)
и
![](data:image/gif;base64,R0lGODlhHQATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABAAWAAsAhQAAAAAAAB0AAAAAMx0AMh0dSAAcSAAzMx00NAAzWh0zWh1GbDMAADMAHTIAMjMzWzNZfzNbgEgcAFozAFszM0YzRltISEhZf11/f0huf2xGHW5bNW5GRn9/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZAQIBwSCwaj8YHcjkEMZ9QwUAYKCqLjEBgISwADg1oAuAEaAAK6HFiOF6NbaHj6FlOhGW1RDgZAyJHFUZZAVMAQQA7)
- его корень, такой что НОД(
r,s)=1, то
![](data:image/gif;base64,R0lGODlhIwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQAfAA8AhAAAAAAAAB0AAB0AMgAAMwAcSAAzWh1GbDMAADMzWzNbgEgcAEgdHVozAEgzM1szM1tISEZGRll/WV1/f113d2xGHX9/XX9uSHd3ZmaIiIBbM4BuboiIZgECAwECAwECAwVJICCOZGme6JmkbOu+5gqbSBDMtsIaqEwKB0ClNYAhgrCCqkR4QUbI1PC1EDWWpKZIiaoFtC5bVCQ5vnysyGytG6fQrcc6LkIoQwA7)
и
![](data:image/gif;base64,R0lGODlhIQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABQAdAA8AhAAAAAAAAB0AAAAAMx0dSAAcSAAyMh00NB0zWgAzWh1GbDIAHTMAADIAMjMzWzNGbjNbgEgcAFozAFozHV1/f0huf2xGHW5bNW5GM39/XX9uSGaIiIBbM4iIZgECAwECAwVNICCOZGmeaOmkbOu+4gqLwiAGcxBAJwEYC5OMJFAALCckAsYwpiSF1FBkYzVmSJMl4GxNAVVAdDQR8WYAXZfHeX3R8BHbDSvHTxIuIAQAOw==)
.