Элементы абстрактной и компьютерной алгебры. Учебное пособие
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§6. ОТДЕЛЕНИЕ КОРНЕЙ
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Ɇɧɨɝɨɱɥɟɧ ± ɷɬɨ ɧɟɧɭɥɟɜɨɣ ɨɫɬɚɬɨɤ ɨɬ ɞɟɥɟɧɢɹ ɦɧɨɝɨɱɥɟɧɚ ɧɚ ɢ ɬ ɞ ɦɧɨɝɨɱɥɟɧ ɪɚɜɟɧ ɧɭɥɸ
ɑɬɨ ɦɨɠɧɨ ɫɤɚɡɚɬɶ ɨ ɦɧɨɝɨɱɥɟɧɟ "
əɫɧɨ ɱɬɨ ɨɧ ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɶɲɢɦ ɨɛɳɢɦ ɞɟɥɢɬɟɥɟɦ ɦɧɨɝɨɱɥɟɧɨɜ ɢ ɬɨ ɟɫɬɶ
ʜʝʓ ǡ ʜʝʓ ǡɉɨɫɤɨɥɶɤɭ ɦɧɨɝɨɱɥɟɧݔɢ ɟɝɨ ɩɪɨɢɡɜɨɞɧɚɹ
ݔɧɟ ɢɦɟɸɬ ɨɛɳɢɯ ɤɨɪɧɟɣ ɬɨ ɦɧɨɝɨɱɥɟɧɵ ɢ ɜɡɚɢɦɧɨ ɩɪɨɫɬɵ ɬɨ ɟɫɬɶ
ɦɧɨɝɨɱɥɟɧ ɢɥɢ ɹɜɥɹɟɬɫɹ ɢɯ ɧɚɢɛɨɥɶɲɢɦ ɨɛɳɢɦ ɞɟɥɢɬɟɥɟɦ ɉɨɷɬɨɦɭ ɛɭɞɟɦ
ɫɱɢɬɚɬɶ ɱɬɨ Ǥ
ɉɨɫɬɪɨɟɧɧɚɹ ɫɢɫɬɟɦɚ ǡ ǡǥǡ ɦɧɨɝɨɱɥɟɧɨɜ ɧɚɡɵɜɚɟɬɫɹɫɢɫɬɟɦɨɣ ɦɧɨɝɨɱɥɟ ɧɨɜ ɒɬɭɪɦɚȻɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɟɟݐȾɜɟ ɩɟɪɜɵɟ ɛɭɤɜɵ ɜ ɨɛɨɡɧɚɱɟɧɢɢ ɜɡɹɬɵ ɢɡ ɮɚɦɢɥɢɢ ɮɪɚɧɰɭɡɫɤɨɝɨ ɦɚɬɟɦɚɬɢɤɚ ɩɨ ɢɦɟɧɢ&KDUOHV )UDQoRLV 6WXUPɚɜɬɨɪɚ ɞɚɧɧɨɝɨ ɦɟɬɨɞɚ
Ⱦɨɝɨɜɨɪɢɦɫɹ ɜɫɟ ɦɧɨɝɨɱɥɟɧɵ ɫɢɫɬɟɦɵ ɦɧɨɝɨɱɥɟɧɚ ɒɬɭɪɦɚ ɤɪɨɦɟ ɩɟɪɜɨɝɨ ɢ ɩɨ ɫɥɟɞɧɟɝɨ ɧɚɡɵɜɚɬɶɩɪɨɦɟɠɭɬɨɱɧɵɦɢ
Ɂɚɦɟɱɚɧɢɟȼ ɞɚɥɶɧɟɣɲɟɦ ɧɚɫ ɛɭɞɭɬ ɢɧɬɟɪɟɫɨɜɚɬɶ ɧɟ ɫɚɦɢ ɡɧɚɱɟɧɢɹ ɦɧɨɝɨɱɥɟ
ɧɨɜ ɫɢɫɬɟɦɵ ɒɬɭɪɦɚ ɜ ɤɚɤɨɣ ɥɢɛɨ ɬɨɱɤɟ ɚ ɬɨɥɶɤɨ ɢɯ ɡɧɚɤɢ ɤɨɬɨɪɵɟ ɧɟ ɦɟɧɹ
ɸɬɫɹ ɩɪɢ ɭɦɧɨɠɟɧɢɢ ɦɧɨɝɨɱɥɟɧɨɜ ɧɚ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɉɨɷɬɨɦɭ
ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɡɚɩɢɫɢ ɦɨɠɧɨ ɜɦɟɫɬɨ ɦɧɨɝɨɱɥɟɧɨɜ ǡǥǡ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɧɨɝɨ
ɱɥɟɧɵ ǡǥǡ ɩɪɢɩɨɥɨɠɢɬɟɥɶɧɵɯ ɦɧɨɠɢɬɟɥɹɯ ǡǥǡ ɢ ɦɵ ɛɭɞɟɦ
ɢɧɨɝɞɚ ɷɬɨ ɞɟɥɚɬɶ
ɁɚɞɚɱɚȾɨɤɚɠɢɬɟ ɱɬɨ ɨɬɫɭɬɫɬɜɢɟ ɤɪɚɬɧɵɯ ɤɨɪɧɟɣ ɭ ɦɧɨɝɨɱɥɟɧɚ ɪɚɜɧɨɫɢɥɶɧɨ
ɨɬɫɭɬɫɬɜɢɸ ɨɛɳɢɯ ɤɨɪɧɟɣ ɭ ɷɬɨɝɨ ɦɧɨɝɨɱɥɟɧɚ ɢ ɟɝɨ ɩɪɨɢɡɜɨɞɧɨɣ
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А.В.Царев,Г.В.Шеина. ЭЛЕМЕНТЫ АБСТРАКТНОЙ И КОМПЬЮТЕРНОЙ АЛГЕБРЫ
ɉɪɨɫɬɟɣɲɢɟ ɫɜɨɣɫɬɜɚ ɫɢɫɬɟɦɵ ɦɧɨɝɨɱɥɟɧɨɜ ɒɬɭɪɦɚ
ɋɜɨɣɫɬɜɨɋɨɫɟɞɧɢɟ ɦɧɨɝɨɱɥɟɧɵ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɦɧɨɝɨɱɥɟɧɨɜ ɒɬɭɪɦɚ
ɧɟ ɢɦɟɸɬ ɨɛɳɢɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɡɚɦɟɬɢɦ ɫɧɚɱɚɥɚ ɱɬɨ ɦɧɨɝɨɱɥɟɧɵ ɢ ɧɟ ɢɦɟɸɬ ɨɛɳɢɯ
ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɫɦ ɡɚɞɚɱɭ ɜɵɲɟ ɉɪɟɞɩɨɥɨɠɢɦ ɩɪɨɬɢɜɧɨɟ ɩɭɫɬɶ ɫɨ
ɫɟɞɧɢɟ ɦɧɨɝɨɱɥɟɧɵ ɢ ɢɦɟɸɬ ɨɛɳɢɣ ɞɟɣɫɬɜɢɬɟɥɶɧɵɣ ɤɨɪɟɧɶ
Ɍɨɝɞɚ Ɋɚɫɫɦɨɬɪɢɦ ɪɚɜɟɧɫɬɜɨ ɢ ɩɨɞ
ɫɬɚɜɢɦ ɜ ɧɟɝɨ
Ɉɱɟɜɢɞɧɨ ɬɟɩɟɪɶ ɱɬɨ Ɋɚɫɫɦɨɬɪɟɜ ɩɚɪɭ ɢ ɚɧɚɥɨɝɢɱɧɵɦ ɨɛ ɪɚɡɨɦ ɩɨɥɭɱɢɦ ɱɬɨ ɉɪɨɞɨɥɠɚɹ ɬɚɤɢɦ ɠɟ ɨɛɪɚɡɨɦ ɩɪɢɯɨɞɢɦ ɤ ɬɨɦɭ
ɱɬɨ ± ɤɨɪɟɧɶ ɜɫɟɯ ɦɧɨɝɨɱɥɟɧɨɜ ǡ ǡǥǡ ǡ ɱɬɨ ɩɪɨɬɢɜɨɪɟɱɢɬ ɬɨɦɭ ɱɬɨ
ɦɧɨɝɨɱɥɟɧɵ ɢ ɧɟ ɢɦɟɸɬ ɨɛɳɢɯ ɤɨɪɧɟɣ ɋɜɨɣɫɬɜɨ ɞɨɤɚɡɚɧɨ
ɁɚɦɟɱɚɧɢɟɆɧɨɝɨɱɥɟɧɵ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢɒɬɭɪɦɚ ɧɟɹɜɥɹɸɳɢɟɫɹɫɨɫɟɞ ɧɢɦɢ ɦɨɝɭɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɨɛɪɚɬɢɬɶɫɹ ɜ ɜ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ ɇɚɩɪɢɦɟɪ ɜ ɤɚɱɟɫɬɜɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɒɬɭɪɦɚ ɞɥɹ ɦɧɨɝɨɱɥɟɧɚݔ ݔ ɦɨɠɧɨ ɜɡɹɬɶ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ݔ ݔ ǡ ݔ ݔ ǡ ݔ ʹ ǡݔ ǡ
Ɇɧɨɝɨɱɥɟɧɵݔ ݔ ݔ ήݔ ɢݔ ɨɛɪɚɳɚɸɬɫɹ ɜ ɩɪɢݔ
ɋɜɨɣɫɬɜɨȿɫɥɢɩɪɨɦɟɠɭɬɨɱɧɵɣɦɧɨɝɨɱɥɟɧ ɢ ɨɛɪɚɳɚɟɬɫɹ
ɜ ɜ ɬɨɱɤɟ ɬɨ ɫɨɫɟɞɧɢɣ ɫɥɟɜɚ ɢ ɫɨɫɟɞɧɢɣ ɫɩɪɚɜɚ ɦɧɨɝɨɱɥɟɧɵ ɢ ɧɟ
ɪɚɜɧɵ ɧɭɥɸ ɜ ɬɨɱɤɟ ɢ ɢɦɟɸɬ ɜ ɬɨɱɤɟ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɡɧɚɤ
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨɩɨɫɜɨɣɫɬɜɭ ɟɫɥɢ ɬɨ ɢ
ɋɧɨɜɚ ɤɚɤ ɢ ɜɵɲɟ ɪɚɫɫɦɨɬɪɢɦ ɪɚɜɟɧɫɬɜɨ
102
§6. ОТДЕЛЕНИЕ КОРНЕЙ
ɢɡ ɤɨɬɨɪɨɝɨ ɩɨɥɭɱɚɟɦ ɋɜɨɣɫɬɜɨ ɞɨɤɚɡɚɧɨ
ɋɱɟɬɱɢɤ ɩɟɪɟɦɟɧ ɡɧɚɤɚ ɢ ɟɝɨ ɫɜɨɣɫɬɜɚ
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɨɫɬɨɹɳɭɸ ɢɡ ɱɢɫɟɥʹǢ Ǣ ͵Ǣ Ǣ Ǣ ʹǢ
ȼ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɡɧɚɤ ɦɟɧɹɟɬɫɹ ɪɚɡɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɝɨ ɱɢɫɥɚ ɤ
ɱɢɫɥɭ ɫɨɫɟɞɧɟɦɭ ɫ ɧɢɦ ɫɩɪɚɜɚ
Ǣ ͵ ͵Ǣ Ǣ ʹǢ
Ȼɭɞɟɦ ɡɚɩɢɫɵɜɚɬɶ ɷɬɨ ɬɚɤݓ ɢ ɧɚɡɵɜɚɬɶ ɮɭɧɤɰɢɸݓ ɫɱɟɬɱɢɤɨɦ ɩɟɪɟɦɟɧ
ɡɧɚɤɚȿɫɥɢ ɜɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɢɦɟɸɬɫɹ ɧɭɥɢɬɨ ɭɞɚɥɢɦ ɢɯ ɢ ɡɚɬɟɦ ɩɨɞ
ɫɱɢɬɚɟɦ ɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɨɜ
ɋɜɨɣɫɬɜɨɑɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɨɜ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɟɫɬɶ ɫɭɦɦɚ ɱɢɫɥɚ ɩɟ
ɪɟɦɟɧ ɡɧɚɤɨɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ ɫɨɫɬɨɹɳɢɯ ɢɡ ɞɜɭɯ ɫɨɫɟɞɧɢɯ ɱɥɟɧɨɜ ɩɨɫɥɟ
ɞɨɜɚɬɟɥɶɧɨɫɬɢ ɬɨ ɟɫɬɶݓɦɨɠɧɨ ɜɵɱɢɫɥɹɬɶ ɬɚɤ
ݓʹǢ ݓǢ ͵ ݓ͵Ǣ ݓǢ ݓǢ ʹ ݓʹǢ
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ɋɜɨɣɫɬɜɨɉɪɢ ɜɵɱɢɫɥɟɧɢɢݓɦɨɠɧɨ ɨɛɴɟɞɢɧɹɬɶ ɧɟɤɨɬɨɪɵɟ ɱɚɫɬɢ ɷɬɨɣ ɩɨ
ɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢɬɚɤɱɬɨɛɵɫɨɫɟɞɧɢɟɮɪɚɝɦɟɧɬɵɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢɢɦɟɥɢ
ɬɨɥɶɤɨ ɨɞɧɭ ɨɛɳɭɸ ɝɪɚɧɢɱɧɭɸ ɬɨɱɤɭ ɇɚɩɪɢɦɟɪ
ݓ ݓʹǢ Ǣ ͵ ݓ͵Ǣ Ǣ ݓǢ ʹǢ
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ɋɜɨɣɫɬɜɨɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ ɟɫɥɢ ɢɦɟɟɬɫɹ ɞɜɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɢ
ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɞɥɢɧɵ ɬɨ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɨɫɬɨ
ɹɳɭɸ ɢɡ ɪɚɡɧɨɫɬɟɣ ɷɥɟɦɟɧɬɨɜ ɷɬɢɯ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ
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А.В.Царев,Г.В.Шеина. ЭЛЕМЕНТЫ АБСТРАКТНОЙ И КОМПЬЮТЕРНОЙ АЛГЕБРЫ
Ɍɨɝɞɚ ɪɚɡɧɨɫɬɶ ݓ ݓ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫɥɨɠɟɧɢɟɦ ɪɚɡɧɨɫɬɟɣ
ݓ ݓ ɝɞɟ ɢ í ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɱɚɫɬɢ ɢɫɯɨɞɧɵɯ ɩɨɫɥɟɞɨɜɚɬɟɥɶ
ɧɨɫɬɟɣ ɢ ɫɨɫɬɚɜɥɟɧɧɵɟ ɩɨɞɨɛɧɨ ɱɚɫɬɹɦ ɢ ɢɡ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɜ
ɫɜɨɣɫɬɜɟ ɩɪɢɦɟɪɚ
ɋɱɟɬɱɢɤ ɩɟɪɟɦɟɧ ɡɧɚɤɨɜ Ɉɫɧɨɜɧɚɹ ɥɟɦɦɚ
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ǡ ǡǥǡ ɡɧɚɱɟɧɢɣ ɫɢɫɬɟɦɵ ɒɬɭɪɦɚ ǡ ǡǥǡ ɦɧɨɝɨɱɥɟɧɚ ɜ ɬɨɱɤɟ Ɉɛɨɡɧɚɱɢɦ ɟɟݐɢ ɪɚɫɫɦɨɬ
ɪɢɦݓ ݐ± ɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɨɜ ɜ ɷɬɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ Ⱦɥɹ ɤɪɚɬ
ɤɨɫɬɢ ɜɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ ݓ ݐ
ȼɟɥɢɱɢɧɭ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɱɢɫɥɚ ɤɨɪɧɟɣ ɦɧɨɝɨɱɥɟɧɚ
ɤɚɤ ɧɚ ɜɫɟɣ ɩɪɹɦɨɣ ɬɚɤ ɢ ɜ ɩɪɨɢɡɜɨɥɶɧɨɦ ɩɨɥɭɢɧɬɟɪɜɚɥɟǢ
Ɉɛɴɹɫɧɢɦ ɤɚɤ ɷɬɨ ɫɞɟɥɚɬɶ Ⱦɥɹ ɷɬɨɝɨ ɪɚɫɫɦɨɬɪɢɦ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɢ
ɅɟɦɦɚɎɭɧɤɰɢɹݔ
xɩɨɫɬɨɹɧɧɚ ɜ ɩɪɨɦɟɠɭɬɤɚɯ ɦɟɠɞɭ ɤɨɪɧɹɦɢ ɦɧɨɝɨɱɥɟɧɚ
xɧɟɩɪɟɪɵɜɧɚ ɫɩɪɚɜɚ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɩɪɹɦɨɣ ɢ ɩɪɢ ɩɟɪɟɯɨɞɟ ɱɟɪɟɡ ɤɨɪɟɧɶ ɦɧɨɝɨɱɥɟɧɚ ɭɦɟɧɶɲɚɟɬ ɫɜɨɟ ɡɧɚɱɟɧɢɟ ɧɚ ɟɞɢɧɢɰɭ
ȾɨɤɚɡɚɬɟɥɶɫɬɜɨɊɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɨ ݔ ǡǥǡݔ ɜɫɟɯɤɨɪɧɟɣɜɫɟɯ ɦɧɨ
ɝɨɱɥɟɧɨɜɫɢɫɬɟɦɵ ɒɬɭɪɦɚ ȼɨɡɧɢɤɚɟɬɫɢɫɬɟɦɚ ɢɧɬɟɪɜɚɥɨɜ
λǢݔ ǡݔ Ǣݔ ǡǥǡݔ Ǣݔ ǡݔ Ǣ λ
ɉɪɢ ɷɬɨɦ ɟɫɥɢ ± ɩɭɫɬɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨ ɪɚɡɛɢɟɧɢɹ ɧɟ ɩɪɨɢɡɨɣɞɟɬ
ɋɪɚɜɧɢɦ ɡɧɚɱɟɧɢɹݔɜ ɪɚɡɥɢɱɧɵɯ ɬɨɱɤɚɯݔ א Ⱦɥɹ ɷɬɨɝɨ ɡɚɦɟɬɢɦ ɱɬɨ ɩɨ
ɬɟɨɪɟɦɟ ȼɟɣɟɪɲɬɪɚɫɫɚ ɜɧɭɬɪɢ ɥɸɛɨɝɨ ɢɧɬɟɪɜɚɥɚ ɧɟ ɩɪɨɢɫɯɨɞɢɬ ɢɡɦɟɧɟɧɢɹ ɡɧɚ
ɤɨɜ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ݔ ǡ ݔ ǡǥǡ ݔ
104
§6. ОТДЕЛЕНИЕ КОРНЕЙ
ȼɵɹɫɧɢɦ ɤɚɤ ɢɡɦɟɧɹɟɬɫɹ ɮɭɧɤɰɢɹ ݔ ɩɪɢ ɩɪɨɯɨɠɞɟɧɢɢ ɱɟɪɟɡ ɤɨɪɟɧɶ
ݔ א ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɦ ɪɚɜɟɧɫɬɜɨ ɞɥɹ ɨɞɧɨɝɨ ɢɥɢ ɧɟɫɤɨɥɶ
ɤɢɯ ɢɧɞɟɤɫɨɜ
Ɋɚɫɫɦɨɬɪɢɦ ɫɧɚɱɚɥɚ ɫɥɭɱɚɣ ɤɨɝɞɚ ǡ ɬɨ ɟɫɬɶ ±ɩɪɨɦɟɠɭɬɨɱɧɵɣ ɦɧɨɝɨɱɥɟɧ ɫɢɫɬɟɦɵ ɒɬɭɪɦɚ Ⱦɥɹ ɷɬɨɝɨ ɜɵɛɟɪɟɦ ɧɟɤɨɬɨɪɭɸ ɨɤɪɟɫɬɧɨɫɬɶǢ
ɬɨɱɤɢ ݔ ɧɟ ɫɨɞɟɪɠɚɳɭɸ ɞɪɭɝɢɯݔ ɢɡ ɦɧɨɠɟɫɬɜɚ
Ɉɩɪɟɞɟɥɢɦ ɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɨɜ ɧɚ ɭɱɚɫɬɤɟ ǡ ǡ ǡ ɜ ɪɚɡ
ɥɢɱɧɵɯ ɬɨɱɤɚɯ ɷɬɨɝɨ ɢɧɬɟɪɜɚɥɚ ɉɨ ɫɜɨɣɫɬɜɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɒɬɭɪɦɚ
ɢɦɟɟɦ ɱɢɫɥɚ ɢ ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ ɢ ɢɦɟɸɬ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɡɧɚɤɢ Ʉɪɨɦɟ ɬɨɝɨ
Ȼɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɨɛɵɱɧɨɟ ɨɛɨɡɧɚɱɟɧɢɟݔɞɥɹ ɡɧɚɤɚ ɱɢɫɥɚݔɢ ɨɛɨɡɧɚɱɢɦ
Ɍɨɝɞɚ ɦɨɠɟɦ ɡɚɩɨɥɧɢɬɶ ɫɥɟɞɭɸɳɭɸ ɬɚɛɥɢɰɭɢɡɤɨɬɨɪɨɣ ɜɢɞɧɨ ɱɬɨ ɧɚ ɭɱɚɫɬɤɟ ǡ ǡ ɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɚɨɞɢɧɚɤɨɜɨɞɥɹ ɜɫɟɯ ɬɨɱɟɤ ɢɧɬɟɪɜɚɥɚǢ
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ݔ ǡ ݔ ǡ ݔ |
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ȼɵɲɟ ɦɵ ɨɬɦɟɱɚɥɢ ɱɬɨݔ ݓ ݐ ݔɦɨɠɧɨ ɜɵɱɢɫɥɹɬɶ ɩɨ ɮɪɚɝɦɟɧɬɚɦ
ɧɚ ɤɨɬɨɪɵɟ ɪɚɡɛɢɬɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɢɫɟɥݐ ݔȿɫɥɢ ɧɚ ɭɱɚɫɬɤɟ
ǡǥǡ ɧɟɬ ɧɭɥɟɣ ɬɨ ɧɚ ɭɱɚɫɬɤɚɯ ǡǥǡ ɢ ǡǥǡ ɡɧɚɤɢ ɛɭɞɭɬ ɫɨɜɩɚɞɚɬɶ ɫɨ ɡɧɚɤɚɦɢ ɱɢɫɟɥ ǡǥǡ
105
А.В.Царев,Г.В.Шеина. ЭЛЕМЕНТЫ АБСТРАКТНОЙ И КОМПЬЮТЕРНОЙ АЛГЕБРЫ
ɉɨɫɥɟɞɧɟɟ ɱɬɨ ɨɫɬɚɟɬɫɹ ɡɚɦɟɬɢɬɶ ± ɷɬɨ ɬɨɬ ɮɚɤɬ ɱɬɨ ɟɫɥɢ ± ɩɟɪɜɨɟ ɧɭɥɟ ɜɨɟ ɡɧɚɱɟɧɢɟ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ǡ ǡǥǡ ɬɨ ɫɥɟɞɭɸɳɟɟ ɧɭɥɟɜɨɟ ɡɧɚɱɟɧɢɟ ɜɫɬɪɟɬɢɬɫɹ ɧɟ ɪɚɧɶɲɟ ɱɟɦ ɞɥɹ
ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɫɭɦɦɢɪɨɜɚɬɶ ɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɨɜ ɨɬɞɟɥɶɧɨ ɩɨ ɭɱɚɫɬɤɭ
ǡ ǡ ɢ ɩɨ ɭɱɚɫɬɤɭ ǡ ǡ ɟɫɥɢ ɢɥɢ ɩɪɨɩɭɫɬɢɬɶ
ɭɱɚɫɬɨɤ ɫ ɧɟɢɡɦɟɧɢɜɲɢɦɢɫɹ ɡɧɚɤɚɦɢ ɟɫɥɢ ɧɨɦɟɪ ɫɥɟɞɭɸɳɟɝɨ ɧɭɥɟɜɨɝɨ
ɡɧɚɱɟɧɢɹ ɬɨ ɟɫɬɶ ɧɚɯɨɞɢɬɫɹ ɞɚɥɶɲɟ ɱɟɦ ɧɨɦɟɪ ʹ
ɉɨɞɜɨɞɹ ɢɬɨɝ ɩɨɥɭɱɚɟɦ ɱɬɨɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɚ ɞɥɹ ɮɪɚɝɦɟɧɬɚ ǡǥǡ ɧɟ ɡɚ ɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɬɨɱɤɢ ɢɡ ɢɧɬɟɪɜɚɥɚ Ǣɢ ɡɧɚɱɢɬ ɹɜɥɹɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɜɟ
ɥɢɱɢɧɨɣ ɞɥɹ ɥɸɛɨɝɨ ɢɧɬɟɪɜɚɥɚݔ Ǣݔ
Ɉɫɬɚɥɨɫɶ ɩɨɧɹɬɶ ɱɬɨ ɩɪɨɢɫɯɨɞɢɬ ɫɨ ɡɧɚɤɚɦɢ ɧɚ ɭɱɚɫɬɤɟ ǡ Ɂɚɦɟɬɢɦ ɱɬɨ ɜ ɫɢɥɭ ɨɬɫɭɬɫɬɜɢɹ ɤɪɚɬɧɵɯ ɤɨɪɧɟɣ ɱɢɫɥɚ ݔɢ ݔ ݔ א ɧɟ ɪɚɜɧɵ ɧɭɥɸ ɨɞ
ɧɨɜɪɟɦɟɧɧɨ ȼɨɡɦɨɠɧɵ ɞɜɚ ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɪɚɡɥɢɱɧɵɯ ɫɥɭɱɚɹ
ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɦɧɨɝɨɱɥɟɧɚ
ɟɫɬɶ ɤɨɪɟɧɶ ɦɧɨɝɨɱɥɟɧɚ
ɉɭɫɬɶ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɵ ɭɠɟ ɜɤɥɸɱɢɥɢ ǡ ɜɨ ɮɪɚɝɦɟɧɬ
ǡ ǡ ɢ ɪɚɫɫɦɨɬɪɟɥɢ ɟɝɨ ɜɵɲɟ Ɍɨ ɠɟ ɤɚɫɚɟɬɫɹ ɫɥɭɱɚɹ ɬɚɤ ɤɚɤ ɬɨɝɞɚ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɡɧɚɤɨɜ ݔ ǡ ݔɢɦɟɟɬ ɨɞɢɧɚɤɨɜɵɣ ɜɢɞ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɢɧɬɟɪɜɚɥɚǢɢ ɡɧɚɱɢɬ ɧɟ ɢɡɦɟɧɹɟɬ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ
ɉɨɞɜɨɞɹ ɢɬɨɝ ɩɨɥɭɱɚɟɦ ɱɬɨ ɮɭɧɤɰɢɹ ݔ ɩɨɫɬɨɹɧɧɚ ɜ ɢɧɬɟɪɜɚɥɟ Ǣ
ɟɫɥɢ ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɦɧɨɝɨɱɥɟɧɚ
Ɉɬɫɸɞɚ ɜ ɱɚɫɬɧɨɫɬɢ ɩɨɥɭɱɚɟɦ ɱɬɨݔ ɧɟ ɢɡɦɟɧɹɟɬ ɫɜɨɟɝɨ ɡɧɚɱɟɧɢɹ ɩɪɢ ɩɟɪɟ
ɯɨɞɟ ɱɟɪɟɡ ɤɨɪɟɧɶ ɦɧɨɝɨɱɥɟɧɚ ɫɢɫɬɟɦɵ ɒɬɭɪɦɚ ɟɫɥɢ ɨɧ ɧɟ ɹɜɥɹɟɬɫɹ ɨɞɧɨɜɪɟ
ɦɟɧɧɨ ɤɨɪɧɟɦ ɢɫɯɨɞɧɨɝɨ ɦɧɨɝɨɱɥɟɧɚ ɢ ɢɦɟɟɬ ɨɞɧɨ ɢ ɬɨ ɠɟ ɡɧɚɱɟɧɢɟ ɜ ɥɸɛɨɦ
ɢɧɬɟɪɜɚɥɟݔ Ǣݔ Ǥ
106
§6. ОТДЕЛЕНИЕ КОРНЕЙ
ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɟɫɬɶ ɤɨɪɟɧɶ ɦɧɨɝɨɱɥɟɧɚ Ɉɩɪɟɞɟɥɢɦ ɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɨɜ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢݐɧɚ ɭɱɚɫɬɤɟ ǡ əɫɧɨ ɱɬɨ ݔɧɟ ɢɦɟɟɬ ɤɨɪɧɟɣ ɜ ɢɧɬɟɪɜɚɥɟǢɩɨɫɤɨɥɶɤɭ ɭ ɦɧɨɝɨɱɥɟɧɨɜ ɢ ɧɟɬ ɨɛɳɢɯ ɤɨɪɧɟɣ
ɂɫɫɥɟɞɭɟɦ ɡɧɚɤ ݔɜ ɢɧɬɟɪɜɚɥɟǢɉɨɫɤɨɥɶɤɭ ɬɨ ɡɧɚɤ ݔɨɞɢ ɧɚɤɨɜ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɢɧɬɟɪɜɚɥɚǢɉɨɷɬɨɦɭ ɜɨɡɦɨɠɧɵ ɞɜɚ ɫɥɭɱɚɹ
ɚ ݔɢɦɟɟɬ ɡɧɚɤ ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɹݔɜɨɡɪɚɫɬɚɟɬ ɜ ɢɧɬɟɪɜɚɥɟǢɢ ɬɨ
ɝɞɚ ɢɦɟɟɦ ɬɚɛɥɢɰɭ
ݔ |
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ɑɢɫɥɨ ɫɦɟɧ ɡɧɚɤɚ |
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ɛ ݔɢɦɟɟɬ ɡɧɚɤ ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɹݔɭɛɵɜɚɟɬ ɜ ɢɧɬɟɪɜɚɥɟǢ ǡ
ɢ ɬɨɝɞɚ ɢɦɟɟɦ ɬɚɛɥɢɰɭ
ݔ |
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ɑɢɫɥɨ ɫɦɟɧ ɡɧɚɤɚ |
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Ɉɛɨɡɧɚɱɢɦ ɱɢɫɥɨ ɩɟɪɟɦɟɧ ɡɧɚɤɚ ɞɥɹ ɮɪɚɝɦɟɧɬɚ ǡǥǡ ɜ ɢɧɬɟɪɜɚɥɟǢɫɢɦ ɜɨɥɨɦ ɋ ɭɱɟɬɨɦ ɬɚɛɥɢɰ ɫɥɭɱɚɟɜ ɚ ɢ ɛ ɩɨɥɭɱɚɟɦ ɱɬɨ ɮɭɧɤɰɢɹݔɜ ɨɤɪɟɫɬɧɨ
ɫɬɢ ɬɨɱɤɢݔ ɹɜɥɹɸɳɟɣɫɹ ɤɨɪɧɟɦ ɦɧɨɝɨɱɥɟɧɚ ɢɦɟɟɬ ɜɢɞ ɢɡɨɛɪɚɠɟɧɧɵɣ
ɧɢɠɟ
107
А.В.Царев,Г.В.Шеина. ЭЛЕМЕНТЫ АБСТРАКТНОЙ И КОМПЬЮТЕРНОЙ АЛГЕБРЫ
Ƚɪɚɮɢɤ ɮɭɧɤɰɢɢݔɜ ɢɧɬɟɪɜɚɥɟǢ
ݕ
ݔ
Ɇɵ ɜɢɞɢɦ ɱɬɨ ɮɭɧɤɰɢɹݔɩɨɫɬɨɹɧɧɚ ɜ ɩɪɨɦɟɠɭɬɤɚɯݔ Ǣݔ ɢ ɧɟɩɪɟɪɵɜɧɚ ɫɩɪɚɜɚ ɉɪɢ ɩɟɪɟɯɨɞɟ ɱɟɪɟɡ ɤɨɪɟɧɶ ɦɧɨɝɨɱɥɟɧɚ ɮɭɧɤɰɢɹݔɭɦɟɧɶɲɚɟɬ ɫɜɨɟ ɡɧɚɱɟɧɢɟ ɧɚ ɟɞɢɧɢɰɭ
ɂɡɨɛɪɚɡɢɦɝɪɚɮɢɤɮɭɧɤɰɢɢ ݔ ɧɚɜɫɟɣɱɢɫɥɨɜɨɣɩɪɹɦɨɣɟɫɥɢ ǡǥǡ Ȃ
ɤɨɪɧɢ ɦɧɨɝɨɱɥɟɧɚ
Ƚɪɚɮɢɤ ɮɭɧɤɰɢɢݔ
ݕ
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Ɏɨɪɦɭɥɢɪɨɜɤɚ ɬɟɨɪɟɦɵ ɒɬɭɪɦɚ ɉɪɢɦɟɪɵ ɢ ɭɩɪɚɠɧɟɧɢɹ
Ɍɟɩɟɪɶ ɦɵ ɦɨɠɟɦ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɭɸ ɬɟɨɪɟɦɭ
Ɍɟɨɪɟɦɚ ɒɬɭɪɦɚɉɭɫɬɶ ɦɧɨɝɨɱɥɟɧ ɧɟ ɢɦɟɟɬ ɤɪɚɬɧɵɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪ
ɧɟɣ ɱɢɫɥɚ ɢ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɩɪɢɱɟɦ Ɍɨɝɞɚ ɪɚɡɧɨɫɬɶ
ɟɫɬɶ ɱɢɫɥɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɦɧɨɝɨɱɥɟɧɚ ɜ ɩɨɥɭɢɧɬɟɪɜɚɥɟǢ
108
§6. ОТДЕЛЕНИЕ КОРНЕЙ
Ɂɚɦɟɱɚɧɢɟɑɚɫɬɨ ɬɟɨɪɟɦɭ ɒɬɭɪɦɚ ɮɨɪɦɭɥɢɪɭɸɬ ɬɚɤ ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɞɜɚ ɞɟɣ
ɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɥɚ ɢ ɩɪɢɱɟɦ ɢ ɧɟ ɹɜɥɹɸɬɫɹ ɤɨɪɧɹɦɢ ɦɧɨɝɨɱɥɟɧɚ
Ɍɨɝɞɚ ɱɢɫɥɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɜ ɢɧɬɟɪɜɚɥɟǢɞɪɭɝɨɣ ɜɚɪɢɚɧɬ ɧɚ ɨɬ
ɪɟɡɤɟǢɪɚɜɧɨ
ɉɪɢɦɟɪɉɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɱɢɫɥɚ ɤɨɪɧɟɣ ɦɧɨɝɨɱɥɟɧɚݔ
ݔ Ʉɚɤ ɨɬɦɟɱɚɥɨɫɶ ɜɵɲɟ ɜ ɤɚɱɟɫɬɜɟ ɫɢɫɬɟɦɵ ɦɧɨɝɨɱɥɟɧɨɜ ɦɨɠɧɨ ɜɡɹɬɶ ɩɨ
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ɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ |
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ɉɨɫɤɨɥɶɤɭʹ ʹ ɬɨ ɦɧɨɝɨɱɥɟɧݔ ݔ ɜ ɩɨɥɭɢɧɬɟɪ
ɜɚɥɟǢʹɢɦɟɟɬ ɨɞɢɧ ɤɨɪɟɧɶ Ɂɚɨɞɧɨ ɜɢɞɢɦ ɱɬɨʹɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ
ɉɪɢɦɟɪɈɩɪɟɞɟɥɢɦ ɱɢɫɥɨ ɤɨɪɧɟɣ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɦɧɨɝɨɱɥɟɧɚ ɧɚ ɜɫɟɣ ɱɢɫ
ɥɨɜɨɣ ɩɪɹɦɨɣ ɉɨɫɤɨɥɶɤɭ ɱɢɫɥɨ ɤɨɪɧɟɣ ɤɨɧɟɱɧɨ ɨɧɢ ɧɚɯɨɞɹɬɫɹ ɜ ɧɟɤɨɬɨɪɨɦ ɢɧ
ɬɟɪɜɚɥɟǢ ǡ א Ǥɇɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɧɚɯɨɞɢɬɶ ɷɬɨ ɱɢɫɥɨ ɉɨɥɶɡɭɸɬɫɹ ɬɟɦ ɱɬɨ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɯ ɡɧɚɱɟɧɢɹɯ ɡɧɚɤ ɦɧɨɝɨɱɥɟɧɚ ɫɨɜɩɚɞɚɟɬ ɫɨ ɡɧɚɤɨɦ ɟɝɨ ɫɬɚɪɲɟɝɨ ɨɞɧɨɱɥɟɧɚ
Ɂɚɩɨɥɧɢɦ ɬɚɛɥɢɰɭ
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А.В.Царев,Г.В.Шеина. ЭЛЕМЕНТЫ АБСТРАКТНОЙ И КОМПЬЮТЕРНОЙ АЛГЕБРЫ
ɂɡ ɬɚɛɥɢɰɵ ɜɢɞɢɦ ɱɬɨ ɨɛɳɟɟ ɱɢɫɥɨ ɤɨɪɧɟɣ ɦɧɨɝɨɱɥɟɧɚݔ ݔ ɪɚɜɧɨ ɟɞɢ
ɧɢɰɟ ɢ ɷɬɨɬ ɤɨɪɟɧɶ ɤɚɤ ɦɵ ɨɛɧɚɪɭɠɢɥɢ ɪɚɧɶɲɟ ɧɚɯɨɞɢɬɫɹ ɜ ɢɧɬɟɪɜɚɥɟǢʹ
ɉɪɢɜɟɞɟɦ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢݕ ݔ ݔ
ɉɪɢɜɟɞɟɧɧɵɣ ɝɪɚɮɢɤ ɩɨɞɬɜɟɪɠɞɚɟɬ ɪɟɡɭɥɶɬɚɬɵ ɧɚɲɟɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ
ɍɩɪɚɠɧɟɧɢɹɋɨɫɬɚɜɶɬɟ ɪɹɞ ɒɬɭɪɦɚ ɢ ɨɩɪɟɞɟɥɢɬɟ ɱɢɫɥɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɦɧɨ
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Ɉɩɪɟɞɟɥɢɬɟ ɱɢɫɥɨ ɤɨɪɧɟɣ ɦɧɨɝɨɱɥɟɧɨɜ ɭɩɪɚɠɧɟɧɢɹ ɚ ± ɟ ɜ ɢɧɬɟɪɜɚɥɚɯ |
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Ɉɬɜɟɬɞɥɹ ɭɩɪɚɠɧɟɧɢɹ |
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Ɉɩɪɟɞɟɥɢɬɟ ɱɢɫɥɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ |
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