원분다항식(cyclotomic polynomial)

수학노트
http://bomber0.myid.net/ (토론)님의 2011년 7월 3일 (일) 12:40 판
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이 항목의 스프링노트 원문주소

 

 

정의
  • \(\Phi_n(X) = \prod_\omega (X-\omega)\)
    • 여기서 \(\omega\)는 primitive n-th root of unity (단위근)
  • 차수는 오일러의 totient 함수 를 사용하여 \(\varphi(n)\) 로 표현됨

 

 

원분다항식 목록
  • 매쓰매티카 Do[Print["\Phi_", i, "(x)=", Cyclotomic[i, x] // TraditionalForm], {i, 1, 20}]

\(\Phi_1(X) = X-1\)

\(\Phi_2(X) = X+1\)

\(\Phi_3(X) = X^2 + X + 1\)

\(\Phi_4(x)=x^2+1\)

\(\Phi_5(x)=x^4+x^3+x^2+x+1\)

\(\Phi_6(X) = X^2 - X + 1\)
\(\Phi_7(x)=x^6+x^5+x^4+x^3+x^2+x+1\)

\(\Phi_8(x)=x^4+1\)

\(\Phi_9(X) = X^6 + X^3 + 1\)
\(\Phi_{10}(x)=x^4-x^3+x^2-x+1\)
\(\Phi_{11}(x)=x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1\)
\(\Phi_{12}(x)=x^4-x^2+1\)
\Phi_13(x)=x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_14(x)=x^6-x^5+x^4-x^3+x^2-x+1

\Phi_{15}(X) = X^8 - X^7 + X^5 - X^4 + X^3 - X + 1
\Phi_16(x)=x^8+1
\Phi_17(x)=x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_18(x)=x^6-x^3+1
\Phi_19(x)=x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_20(x)=x^8-x^6+x^4-x^2+1
\Phi_21(x)=x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1
\Phi_22(x)=x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_23(x)=x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_24(x)=x^8-x^4+1
\Phi_25(x)=x^20+x^15+x^10+x^5+1
\Phi_26(x)=x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_27(x)=x^18+x^9+1
\Phi_28(x)=x^12-x^10+x^8-x^6+x^4-x^2+1
\Phi_29(x)=x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_30(x)=x^8+x^7-x^5-x^4-x^3+x+1
\Phi_31(x)=x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_32(x)=x^16+1
\Phi_33(x)=x^20-x^19+x^17-x^16+x^14-x^13+x^11-x^10+x^9-x^7+x^6-x^4+x^3-x+1
\Phi_34(x)=x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_35(x)=x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1
\Phi_36(x)=x^12-x^6+1
\Phi_37(x)=x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_38(x)=x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_39(x)=x^24-x^23+x^21-x^20+x^18-x^17+x^15-x^14+x^12-x^10+x^9-x^7+x^6-x^4+x^3-x+1
\Phi_40(x)=x^16-x^12+x^8-x^4+1
\Phi_41(x)=x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_42(x)=x^12+x^11-x^9-x^8+x^6-x^4-x^3+x+1
\Phi_43(x)=x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_44(x)=x^20-x^18+x^16-x^14+x^12-x^10+x^8-x^6+x^4-x^2+1
\Phi_45(x)=x^24-x^21+x^15-x^12+x^9-x^3+1
\Phi_46(x)=x^22-x^21+x^20-x^19+x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_47(x)=x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_48(x)=x^16-x^8+1
\Phi_49(x)=x^42+x^35+x^28+x^21+x^14+x^7+1
\Phi_50(x)=x^20-x^15+x^10-x^5+1
\Phi_51(x)=x^32-x^31+x^29-x^28+x^26-x^25+x^23-x^22+x^20-x^19+x^17-x^16+x^15-x^13+x^12-x^10+x^9-x^7+x^6-x^4+x^3-x+1
\Phi_52(x)=x^24-x^22+x^20-x^18+x^16-x^14+x^12-x^10+x^8-x^6+x^4-x^2+1
\Phi_53(x)=x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_54(x)=x^18-x^9+1
\Phi_55(x)=x^40-x^39+x^35-x^34+x^30-x^28+x^25-x^23+x^20-x^17+x^15-x^12+x^10-x^6+x^5-x+1
\Phi_56(x)=x^24-x^20+x^16-x^12+x^8-x^4+1
\Phi_57(x)=x^36-x^35+x^33-x^32+x^30-x^29+x^27-x^26+x^24-x^23+x^21-x^20+x^18-x^16+x^15-x^13+x^12-x^10+x^9-x^7+x^6-x^4+x^3-x+1
\Phi_58(x)=x^28-x^27+x^26-x^25+x^24-x^23+x^22-x^21+x^20-x^19+x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_59(x)=x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_60(x)=x^16+x^14-x^10-x^8-x^6+x^2+1
\Phi_61(x)=x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_62(x)=x^30-x^29+x^28-x^27+x^26-x^25+x^24-x^23+x^22-x^21+x^20-x^19+x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_63(x)=x^36-x^33+x^27-x^24+x^18-x^12+x^9-x^3+1
\Phi_64(x)=x^32+1
\Phi_65(x)=x^48-x^47+x^43-x^42+x^38-x^37+x^35-x^34+x^33-x^32+x^30-x^29+x^28-x^27+x^25-x^24+x^23-x^21+x^20-x^19+x^18-x^16+x^15-x^14+x^13-x^11+x^10-x^6+x^5-x+1
\Phi_66(x)=x^20+x^19-x^17-x^16+x^14+x^13-x^11-x^10-x^9+x^7+x^6-x^4-x^3+x+1
\Phi_67(x)=x^66+x^65+x^64+x^63+x^62+x^61+x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_68(x)=x^32-x^30+x^28-x^26+x^24-x^22+x^20-x^18+x^16-x^14+x^12-x^10+x^8-x^6+x^4-x^2+1
\Phi_69(x)=x^44-x^43+x^41-x^40+x^38-x^37+x^35-x^34+x^32-x^31+x^29-x^28+x^26-x^25+x^23-x^22+x^21-x^19+x^18-x^16+x^15-x^13+x^12-x^10+x^9-x^7+x^6-x^4+x^3-x+1
\Phi_70(x)=x^24+x^23-x^19-x^18-x^17-x^16+x^14+x^13+x^12+x^11+x^10-x^8-x^7-x^6-x^5+x+1
\Phi_71(x)=x^70+x^69+x^68+x^67+x^66+x^65+x^64+x^63+x^62+x^61+x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_72(x)=x^24-x^12+1
\Phi_73(x)=x^72+x^71+x^70+x^69+x^68+x^67+x^66+x^65+x^64+x^63+x^62+x^61+x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_74(x)=x^36-x^35+x^34-x^33+x^32-x^31+x^30-x^29+x^28-x^27+x^26-x^25+x^24-x^23+x^22-x^21+x^20-x^19+x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_75(x)=x^40-x^35+x^25-x^20+x^15-x^5+1
\Phi_76(x)=x^36-x^34+x^32-x^30+x^28-x^26+x^24-x^22+x^20-x^18+x^16-x^14+x^12-x^10+x^8-x^6+x^4-x^2+1
\Phi_77(x)=x^60-x^59+x^53-x^52+x^49-x^48+x^46-x^45+x^42-x^41+x^39-x^37+x^35-x^34+x^32-x^30+x^28-x^26+x^25-x^23+x^21-x^19+x^18-x^15+x^14-x^12+x^11-x^8+x^7-x+1
\Phi_78(x)=x^24+x^23-x^21-x^20+x^18+x^17-x^15-x^14+x^12-x^10-x^9+x^7+x^6-x^4-x^3+x+1
\Phi_79(x)=x^78+x^77+x^76+x^75+x^74+x^73+x^72+x^71+x^70+x^69+x^68+x^67+x^66+x^65+x^64+x^63+x^62+x^61+x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_80(x)=x^32-x^24+x^16-x^8+1
\Phi_81(x)=x^54+x^27+1
\Phi_82(x)=x^40-x^39+x^38-x^37+x^36-x^35+x^34-x^33+x^32-x^31+x^30-x^29+x^28-x^27+x^26-x^25+x^24-x^23+x^22-x^21+x^20-x^19+x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_83(x)=x^82+x^81+x^80+x^79+x^78+x^77+x^76+x^75+x^74+x^73+x^72+x^71+x^70+x^69+x^68+x^67+x^66+x^65+x^64+x^63+x^62+x^61+x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_84(x)=x^24+x^22-x^18-x^16+x^12-x^8-x^6+x^2+1
\Phi_85(x)=x^64-x^63+x^59-x^58+x^54-x^53+x^49-x^48+x^47-x^46+x^44-x^43+x^42-x^41+x^39-x^38+x^37-x^36+x^34-x^33+x^32-x^31+x^30-x^28+x^27-x^26+x^25-x^23+x^22-x^21+x^20-x^18+x^17-x^16+x^15-x^11+x^10-x^6+x^5-x+1
\Phi_86(x)=x^42-x^41+x^40-x^39+x^38-x^37+x^36-x^35+x^34-x^33+x^32-x^31+x^30-x^29+x^28-x^27+x^26-x^25+x^24-x^23+x^22-x^21+x^20-x^19+x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_87(x)=x^56-x^55+x^53-x^52+x^50-x^49+x^47-x^46+x^44-x^43+x^41-x^40+x^38-x^37+x^35-x^34+x^32-x^31+x^29-x^28+x^27-x^25+x^24-x^22+x^21-x^19+x^18-x^16+x^15-x^13+x^12-x^10+x^9-x^7+x^6-x^4+x^3-x+1
\Phi_88(x)=x^40-x^36+x^32-x^28+x^24-x^20+x^16-x^12+x^8-x^4+1
\Phi_89(x)=x^88+x^87+x^86+x^85+x^84+x^83+x^82+x^81+x^80+x^79+x^78+x^77+x^76+x^75+x^74+x^73+x^72+x^71+x^70+x^69+x^68+x^67+x^66+x^65+x^64+x^63+x^62+x^61+x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_90(x)=x^24+x^21-x^15-x^12-x^9+x^3+1
\Phi_91(x)=x^72-x^71+x^65-x^64+x^59-x^57+x^52-x^50+x^46-x^43+x^39-x^36+x^33-x^29+x^26-x^22+x^20-x^15+x^13-x^8+x^7-x+1
\Phi_92(x)=x^44-x^42+x^40-x^38+x^36-x^34+x^32-x^30+x^28-x^26+x^24-x^22+x^20-x^18+x^16-x^14+x^12-x^10+x^8-x^6+x^4-x^2+1
\Phi_93(x)=x^60-x^59+x^57-x^56+x^54-x^53+x^51-x^50+x^48-x^47+x^45-x^44+x^42-x^41+x^39-x^38+x^36-x^35+x^33-x^32+x^30-x^28+x^27-x^25+x^24-x^22+x^21-x^19+x^18-x^16+x^15-x^13+x^12-x^10+x^9-x^7+x^6-x^4+x^3-x+1
\Phi_94(x)=x^46-x^45+x^44-x^43+x^42-x^41+x^40-x^39+x^38-x^37+x^36-x^35+x^34-x^33+x^32-x^31+x^30-x^29+x^28-x^27+x^26-x^25+x^24-x^23+x^22-x^21+x^20-x^19+x^18-x^17+x^16-x^15+x^14-x^13+x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
\Phi_95(x)=x^72-x^71+x^67-x^66+x^62-x^61+x^57-x^56+x^53-x^51+x^48-x^46+x^43-x^41+x^38-x^36+x^34-x^31+x^29-x^26+x^24-x^21+x^19-x^16+x^15-x^11+x^10-x^6+x^5-x+1
\Phi_96(x)=x^32-x^16+1
\Phi_97(x)=x^96+x^95+x^94+x^93+x^92+x^91+x^90+x^89+x^88+x^87+x^86+x^85+x^84+x^83+x^82+x^81+x^80+x^79+x^78+x^77+x^76+x^75+x^74+x^73+x^72+x^71+x^70+x^69+x^68+x^67+x^66+x^65+x^64+x^63+x^62+x^61+x^60+x^59+x^58+x^57+x^56+x^55+x^54+x^53+x^52+x^51+x^50+x^49+x^48+x^47+x^46+x^45+x^44+x^43+x^42+x^41+x^40+x^39+x^38+x^37+x^36+x^35+x^34+x^33+x^32+x^31+x^30+x^29+x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
\Phi_98(x)=x^42-x^35+x^28-x^21+x^14-x^7+1
\Phi_99(x)=x^60-x^57+x^51-x^48+x^42-x^39+x^33-x^30+x^27-x^21+x^18-x^12+x^9-x^3+1
\Phi_100(x)=x^40-x^30+x^20-x^10+1

 

 

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사전형태의 참고자료
원본 주소 "https://wiki.mathnt.net/index.php?title=원분다항식(cyclotomic_polynomial)&oldid=16156"