"원분다항식(cyclotomic polynomial)"의 두 판 사이의 차이

2021년 2월 17일 (수) 04:55 기준 최신판

개요

원분체 (cyclotomic field) 의 연구에서 다룰 수 있는 주요 대상
방정식과 근의 공식 연구의 중요한 실험장

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
\(x^n-1= \prod_{d|n}\Phi_d(x)\)

원분다항식의 상호법칙

소수 \(p\) 에 대해 \(\Phi_n(x) \pmod p\) 가 어떻게 분해되는가의 문제

정리

\(p\in (\mathbb{Z}/n\mathbb{Z})^\times\)의 order가 \(r\)이라 하자. 즉 \(r\)이 \(p^r=1\pmod n\) 을 만족시키는 가장 작은 자연수라 하자.

그러면 \(\Phi_n(x) \pmod p\) 는 차수가 \(r\)인 기약다항식들의 곱으로 표현된다. 즉 \(\Phi_n(x) \pmod p\)의 분해는, \(p\pmod n\)에 의해 결정된다.

증명은 정수론에서의 상호법칙 (reciprocity laws) 참조

따름정리

\(n | p-1\) \(\iff\) \(\Phi_n(x) \pmod p\)는 일차식들로 분해된다

원분다항식 목록

\(\begin{array}{l|l|l} n & \varphi (n) & \Phi _n(x) \\ \hline 1 & 1 & 1-x \\ 2 & 1 & 1+x \\ 3 & 2 & 1+x+x^2 \\ 4 & 2 & 1+x^2 \\ 5 & 4 & 1+x+x^2+x^3+x^4 \\ 6 & 2 & 1-x+x^2 \\ 7 & 6 & 1+x+x^2+x^3+x^4+x^5+x^6 \\ 8 & 4 & 1+x^4 \\ 9 & 6 & 1+x^3+x^6 \\ 10 & 4 & 1-x+x^2-x^3+x^4 \\ 11 & 10 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10} \\ 12 & 4 & 1-x^2+x^4 \\ 13 & 12 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12} \\ 14 & 6 & 1-x+x^2-x^3+x^4-x^5+x^6 \\ 15 & 8 & 1-x+x^3-x^4+x^5-x^7+x^8 \\ 16 & 8 & 1+x^8 \\ 17 & 16 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}+x^{13}+x^{14}+x^{15}+x^{16} \\ 18 & 6 & 1-x^3+x^6 \\ 19 & 18 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}+x^{13}+x^{14}+x^{15}+x^{16}+x^{17}+x^{18} \\ 20 & 8 & 1-x^2+x^4-x^6+x^8 \end{array}\)

\(n=105\)일 때, 0또는 \(\pm 1\)외의 계수가 등장한다

\[ \begin{align*} \Phi_{105}(x)&= 1 + x + x^{2} - x^{5} - x^{6} - 2 x^{7} \\ & \quad -x^{8} - x^{9} + x^{12} + x^{13} + x^{14} + x^{15} \\ & \quad +x^{16} + x^{17} - x^{20} - x^{22} - x^{24} - x^{26} \\ & \quad -x^{28} + x^{31} + x^{32} + x^{33} + x^{34} + x^{35} \\ & \quad +x^{36} - x^{39} - x^{40} - 2 x^{41} - x^{42} - x^{43} \end{align*} \]

@@ 1번째 줄: / 1번째 줄: @@
-<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소</h5>
+==개요==
-* [[원분다항식(cyclotomic polynomial)]]
+* [[원분체 (cyclotomic field)]] 의 연구에서 다룰 수 있는 주요 대상
+* [[방정식과 근의 공식]] 연구의 중요한 실험장
-<h5>정의</h5>
+==정의==
-* <math>\Phi_n(X) = \prod_\omega (X-\omega)</math><br>
+* <math>\Phi_n(X) = \prod_\omega (X-\omega)</math>
 ** 여기서 <math>\omega</math>는 primitive n-th root of unity (단위근)
+* 차수는 [[오일러의 totient 함수]] 를 사용하여 <math>\varphi(n)</math> 로 표현됨
+* <math>x^n-1= \prod_{d|n}\Phi_d(x)</math>
-<h5>100까지의 리스트</h5>
+==원분다항식의 상호법칙==
+* 소수 <math>p</math> 에 대해 <math>\Phi_n(x) \pmod p</math> 가 어떻게 분해되는가의 문제
-* 매쓰매티카 Do[Print["\Phi_", i, "(x)=", Cyclotomic[i, x] // TraditionalForm], {i, 1, 20}]
+;정리
-<math>\Phi_1(X) = X-1</math>
+<math>p\in (\mathbb{Z}/n\mathbb{Z})^\times</math>의 order가 <math>r</math>이라 하자. 즉 <math>r</math>이 <math>p^r=1\pmod n</math> 을 만족시키는 가장 작은 자연수라 하자.
-<math>\Phi_2(X) = X+1</math>
+그러면 <math>\Phi_n(x) \pmod p</math> 는 차수가 <math>r</math>인 기약다항식들의 곱으로 표현된다. 즉 <math>\Phi_n(x) \pmod p</math>의 분해는, <math>p\pmod n</math>에 의해 결정된다.
-<math>\Phi_3(X) = X^2 + X + 1</math>
+* 증명은 [[정수론에서의 상호법칙 (reciprocity laws)]] 참조
-<math>\Phi_4(x)=x^2+1</math>
+;따름정리
+<math>n | p-1</math>  <math>\iff</math>  <math>\Phi_n(x) \pmod p</math>는 일차식들로 분해된다
-<math>\Phi_5(x)=x^4+x^3+x^2+x+1</math>
-<math>\Phi_6(X) = X^2 - X + 1</math><br><math>\Phi_7(x)=x^6+x^5+x^4+x^3+x^2+x+1</math>
-<math>\Phi_8(x)=x^4+1</math>
+==원분다항식 목록==
-<math>\Phi_9(X) = X^6 + X^3 + 1</math><br><math>\Phi_{10}(x)=x^4-x^3+x^2-x+1</math><br><math>\Phi_{11}(x)=x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1</math><br><math>\Phi_{12}(x)=x^4-x^2+1</math><br> \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<br> \Phi_14(x)=x^6-x^5+x^4-x^3+x^2-x+1
+<math>\begin{array}{l|l|l} n  &  \varphi (n) & \Phi _n(x) \\ \hline  1 & 1 & 1-x \\  2 & 1 & 1+x \\  3 & 2 & 1+x+x^2 \\  4 & 2 & 1+x^2 \\  5 & 4 & 1+x+x^2+x^3+x^4 \\  6 & 2 & 1-x+x^2 \\  7 & 6 & 1+x+x^2+x^3+x^4+x^5+x^6 \\  8 & 4 & 1+x^4 \\  9 & 6 & 1+x^3+x^6 \\  10 & 4 & 1-x+x^2-x^3+x^4 \\  11 & 10 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10} \\  12 & 4 & 1-x^2+x^4 \\  13 & 12 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12} \\  14 & 6 & 1-x+x^2-x^3+x^4-x^5+x^6 \\  15 & 8 & 1-x+x^3-x^4+x^5-x^7+x^8 \\  16 & 8 & 1+x^8 \\  17 & 16 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}+x^{13}+x^{14}+x^{15}+x^{16} \\  18 & 6 & 1-x^3+x^6 \\  19 & 18 & 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}+x^{13}+x^{14}+x^{15}+x^{16}+x^{17}+x^{18} \\  20 & 8 & 1-x^2+x^4-x^6+x^8 \end{array}</math>
+* <math>n=105</math>일 때, 0또는 <math>\pm 1</math>외의 계수가 등장한다
+:<math>
+\begin{align*}
+\Phi_{105}(x)&=
++ x + x^{2} - x^{5} - x^{6} - 2 x^{7} \\
+& \quad -x^{8} - x^{9} + x^{12} + x^{13} + x^{14} + x^{15}
+\\
+& \quad +x^{16} + x^{17} - x^{20} - x^{22} - x^{24} - x^{26}
+\\
+& \quad -x^{28} + x^{31} + x^{32} + x^{33} + x^{34} + x^{35}
+\\
+& \quad +x^{36} - x^{39} - x^{40} - 2 x^{41} - x^{42} - x^{43}
+\end{align*}
+</math>
-<math>\Phi_{15}(X) = X^8 - X^7 + X^5 - X^4 + X^3 - X + 1</math><br> \Phi_16(x)=x^8+1<br> \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<br> \Phi_18(x)=x^6-x^3+1<br> \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<br> \Phi_20(x)=x^8-x^6+x^4-x^2+1<br> \Phi_21(x)=x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1<br> \Phi_22(x)=x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1<br> \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<br> \Phi_24(x)=x^8-x^4+1<br> \Phi_25(x)=x^20+x^15+x^10+x^5+1<br> \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<br> \Phi_27(x)=x^18+x^9+1<br> \Phi_28(x)=x^12-x^10+x^8-x^6+x^4-x^2+1<br> \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<br> \Phi_30(x)=x^8+x^7-x^5-x^4-x^3+x+1<br> \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<br> \Phi_32(x)=x^16+1<br> \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<br> \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<br> \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<br> \Phi_36(x)=x^12-x^6+1<br> \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<br> \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<br> \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<br> \Phi_40(x)=x^16-x^12+x^8-x^4+1<br> \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<br> \Phi_42(x)=x^12+x^11-x^9-x^8+x^6-x^4-x^3+x+1<br> \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<br> \Phi_44(x)=x^20-x^18+x^16-x^14+x^12-x^10+x^8-x^6+x^4-x^2+1<br> \Phi_45(x)=x^24-x^21+x^15-x^12+x^9-x^3+1<br> \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<br> \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<br> \Phi_48(x)=x^16-x^8+1<br> \Phi_49(x)=x^42+x^35+x^28+x^21+x^14+x^7+1<br> \Phi_50(x)=x^20-x^15+x^10-x^5+1<br> \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<br> \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<br> \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<br> \Phi_54(x)=x^18-x^9+1<br> \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<br> \Phi_56(x)=x^24-x^20+x^16-x^12+x^8-x^4+1<br> \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<br> \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<br> \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<br> \Phi_60(x)=x^16+x^14-x^10-x^8-x^6+x^2+1<br> \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<br> \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<br> \Phi_63(x)=x^36-x^33+x^27-x^24+x^18-x^12+x^9-x^3+1<br> \Phi_64(x)=x^32+1<br> \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<br> \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<br> \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<br> \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<br> \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<br> \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<br> \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<br> \Phi_72(x)=x^24-x^12+1<br> \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<br> \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<br> \Phi_75(x)=x^40-x^35+x^25-x^20+x^15-x^5+1<br> \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<br> \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<br> \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<br> \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<br> \Phi_80(x)=x^32-x^24+x^16-x^8+1<br> \Phi_81(x)=x^54+x^27+1<br> \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<br> \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<br> \Phi_84(x)=x^24+x^22-x^18-x^16+x^12-x^8-x^6+x^2+1<br> \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<br> \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<br> \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<br> \Phi_88(x)=x^40-x^36+x^32-x^28+x^24-x^20+x^16-x^12+x^8-x^4+1<br> \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<br> \Phi_90(x)=x^24+x^21-x^15-x^12-x^9+x^3+1<br> \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<br> \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<br> \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<br> \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<br> \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<br> \Phi_96(x)=x^32-x^16+1<br> \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<br> \Phi_98(x)=x^42-x^35+x^28-x^21+x^14-x^7+1<br> \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<br> \Phi_100(x)=x^40-x^30+x^20-x^10+1
+==역사==
+* http://functions.wolfram.com/Polynomials/Cyclotomic/35/ShowAll.html
+* [[수학사 연표]]
-<h5>역사</h5>
-* [[수학사연표 (역사)|수학사연표]]
+==관련된 항목들==
+* [[오일러의 totient 함수]]
+* [[가우스와 정17각형의 작도]]
-<h5>상위 주제</h5>
+* [[삼각함수의 값]]
-<h5>재미있는 사실</h5>
+==수학용어번역==
+* {{학술용어집|url=cyclotomic}}
-<h5>관련된 다른 주제들</h5>
+==매스매티카 파일 및 계산 리소스==
-*  [[#|오일러의 totient 함수]]
+* https://docs.google.com/leaf?id=0B8XXo8Tve1cxNWJiOTZkZTYtMDJhMS00MDg4LTljMzItNWFhYjg3MzMwNDRl&sort=name&layout=list&num=50
-* [[가우스와 정17각형의 작도]]
+* http://www.wolframalpha.com/input/?i=cyclotomic+polynomial
+==사전형태의 참고자료==
-<h5>관련도서 및 추천도서</h5>
-*  도서내검색<br>
-** http://books.google.com/books?q=
-** http://book.daum.net/search/contentSearch.do?query=
-*  도서검색<br>
-** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
-** http://book.daum.net/search/mainSearch.do?query=
-<h5>사전형태의 참고자료</h5>
 * http://ko.wikipedia.org/wiki/
 * http://en.wikipedia.org/wiki/Cyclotomic_polynomial
-* http://en.wikipedia.org/wiki/
-* http://www88.wolframalpha.com/input/?i=cyclotomic+polynomial
-* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
-** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=cyclotomic
-* <br>
-<h5>관련기사</h5>
-*  네이버 뉴스 검색 (키워드 수정)<br>
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-<h5>블로그</h5>
+==관련논문==
+* Bartlomiej Bzdega, Products of cyclotomic polynomials on unit circle, arXiv:1606.07622 [math.NT], June 24 2016, http://arxiv.org/abs/1606.07622
+* Pomerance, Carl, Lola Thompson, and Andreas Weingartner. “On Integers <math>n</math> for Which <math>X^n-1</math> Has a Divisor of Every Degree.” arXiv:1511.03357 [math], November 10, 2015. http://arxiv.org/abs/1511.03357.
+* Somu, Sai Teja. “On the Distribution of Numbers Related to the Divisors of <math>x^n-1</math>.” arXiv:1511.03230 [math], November 10, 2015. http://arxiv.org/abs/1511.03230.
+* Somu, Sai Teja. “On the Coefficients of Divisors of X^n-1.” arXiv:1511.03226 [math], November 10, 2015. http://arxiv.org/abs/1511.03226.
+* Damianou, Pantelis A. ‘Monic Polynomials in <math>Z[x]</math> with Roots in the Unit Disc’. arXiv:1507.02419 [math], 9 July 2015. http://arxiv.org/abs/1507.02419.
+* Martínez, F. E. Brochero, C. R. Giraldo Vergara, and L. Batista de Oliveira. “Explicit Factorization of <math>x^n-1\in \mathbb F_q[x]</math>.” arXiv:1404.6281 [cs, Math], April 24, 2014. http://arxiv.org/abs/1404.6281.
-* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
+==메타데이터==
-* 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q1051983 Q1051983]
+===Spacy 패턴 목록===
+* [{'LOWER': 'cyclotomic'}, {'LEMMA': 'polynomial'}]

"원분다항식(cyclotomic polynomial)"의 두 판 사이의 차이

2021년 2월 17일 (수) 04:55 기준 최신판

목차

개요

정의

원분다항식의 상호법칙

원분다항식 목록

역사

관련된 항목들

수학용어번역

매스매티카 파일 및 계산 리소스

사전형태의 참고자료

관련논문

메타데이터

위키데이터

Spacy 패턴 목록

둘러보기 메뉴

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