Cyclotomic polynomial

In mathematics, more specifically in algebra, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients, which is a divisor of $x^n-1$ and is not a divisor of $x^k-1$ for any k < n. Its roots are the nth primitive roots of unity $e^{2i\pi\frac{k}{n}}$ , where k runs over the integers lower than n and coprime to n. In other words, the nth cyclotomic polynomial is equal to

\Phi_n(x) = \prod_\stackrel{1\le k\le n}{\gcd(k,n)=1} \left(x-e^{2i\pi\frac{k}{n}}\right)

It may also be defined as the monic polynomial with integer coefficients, which is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity ( $e^{2i\pi/n}$ is an example of such a root).

Another important equation, linking the cyclotomic polynomials and primitive roots of unity, is the following one.

x^n - 1 =\prod_{1\leq k\leq n} \left(x- e^{2 \pi \cdot \mathrm i k / n} \right)= \prod_{d \mid n} \prod_{1 \leq k \leq n\atop \operatorname{gcd}(k, n) = d} \left(x- e^{2 \pi \cdot \mathrm i k / n} \right) =\prod_{d \mid n} \Phi_{n/d}(x) = \prod_{d\mid n} \Phi_d(x)

Examples

If n is a prime number then

~\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1} x^i.

If n=2p where p is an odd prime number then

~\Phi_{2p}(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{i=0}^{p-1} (-x)^i.

For n up to 30, the cyclotomic polynomials are:^[1]

~\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

\begin{align} \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 \end{align}

~\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

\begin{align} \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 \end{align}

~\Phi_{30}(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

The case of the 105th cyclotomic polynomial is interesting because 105 is the lowest integer that is the product of three distinct odd prime numbers and this polynomial is the first one that has a coefficient other than 1, 0, or -1:

\begin{align} \Phi_{105}(x) = & \; x^{48} + x^{47} + x^{46} - x^{43} - x^{42} - 2 x^{41} - x^{40} - x^{39} + x^{36} + x^{35} + x^{34} \\ & {} + x^{33} + x^{32} + x^{31} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} \\ & {} + x^{14} + x^{13} + x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1 \end{align}

Properties

Fundamental tools

The cyclotomic polynomials are monic polynomials with integer coefficients that are irreducible over the field of the rational numbers. Except for n equal to 1 or 2, they are palindromics of even degree.

The degree of $\Phi_n$ , or in other words the number of nth primitive roots of unity, is $\varphi (n)$ , where $\varphi$ is Euler's totient function.

The fact that $\Phi_n$ is an irreducible polynomial of degree $\varphi (n)$ in the ring $\mathbb{Z}[x]$ is a nontrivial result due to Gauss.^[2] Depending on the chosen definition, it is either the value of the degree or the irreducibility which is a nontrivial result. The case of prime n is easier to prove than the general case, thanks to Eisenstein's criterion.

A fundamental relation involving cyclotomic polynomials is

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

which means that each n-th root of unity is a primitive d-th root of unity for a unique d dividing n.

The Möbius inversion formula allows the expression of $\Phi_n(x)$ as an explicit rational fraction:

\Phi_n(x)=\prod_{d\mid n}(x^d-1)^{\mu(n/d)},

where $\mu$ is the Möbius function.

The Fourier transform of functions of the greatest common divisor together with the Möbius inversion formula gives:^[3]

~\Phi_n(x)=\prod\limits_{k=1}^n{(x^{\operatorname{gcd}(k,n)}-1)^{\cos (\frac{2\pi k}{n})}}

The cyclotomic polynomial $\Phi_{n}(x)$ may be computed by (exactly) dividing $x^n-1$ by the cyclotomic polynomials of the proper divisors of n previously computed recursively by the same method:

\Phi_n(x)=\frac{x^{n}-1}{\prod_{\stackrel{d|n}{{}_{d<n}}}\Phi_{d}(x)}

(Recall that $\Phi_{1}(x)=x-1$ .)

This formula allows to compute $\Phi_n(x)$ on a computer for any n, as soon as integer factorization and division of polynomials are available. Many computer algebra systems have a built in function to compute the cyclotomic polynomials. For example, this function is called by typing cyclotomic_polynomial(n,'x') in Sage, numtheory[cyclotomic](n,x); in Maple, and Cyclotomic[n,x] in Mathematica.

Easy cases for the computation

As noted above, if n is a prime number, then

\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1}x^i.

If n is an odd integer greater than one, then

\Phi_{2n}(x) = \Phi_n(-x).

In particular, if n=2p is twice an odd prime, then (as noted above)

\Phi_n(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{i=0}^{p-1}(-x)^i.

If n=p^m is a prime power (where p is prime), then

\Phi_n(x) = \Phi_p(x^{p^{m-1}}) =\sum_{i=0}^{p-1}x^{ip^{m-1}}.

More generally, if n=p^mr with r relatively prime to p, then

\Phi_n(x) = \Phi_{pr}(x^{p^{m-1}}).

These formulas may be applied repeatedly to get a simple expression for any cyclotomic polynomial $\Phi_n(x)$ in term of a cyclotomic polynomial of square free index: If q is the product of the prime divisors of n (its radical), then^[4]

\Phi_n(x) = \Phi_q(x^{n/q}).

This allows to give formulas for the nth cyclotomic polynomial when n has at most one odd prime factor: If p is an odd prime number, and h and k are positive integers, then:

\Phi_{2^h}(x) = x^{2^{h-1}}+1

\Phi_{p^k}(x) = \sum_{i=0}^{p-1}x^{ip^{k-1}}

\Phi_{2^hp^k}(x) = \sum_{i=0}^{p-1}(-1)^ix^{i2^{h-1}p^{k-1}}

For the other values of n, the computation of the nth cyclotomic polynomial is similarly reduced to that of $\Phi_q(x),$ where q is the product of the distinct odd prime divisors of n. To deal with this case, one has that, for p prime and not dividing n,^[5]

\Phi_{np}(x)=\Phi_{n}(x^p)/\Phi_n(x)\,.

Integers appearing as coefficients

The problem of bounding the magnitude of the coefficients of the cyclotomic polynomials has been the object of a number of research papers.

If n has at most two distinct odd prime factors, then Migotti showed that the coefficients of $\Phi_n$ are all in the set {1, −1, 0}.^[6]

The first cyclotomic polynomial for a product of 3 different odd prime factors is $\Phi_{105}(x);$ it has a coefficient −2 (see its expression above). The converse isn't true: $\Phi_{231}(x)$ = $\Phi_{3\times 7\times 11}(x)$ only has coefficients in {1, −1, 0}.

If n is a product of more odd different prime factors, the coefficients may increase to very high values. E.g., $\Phi_{15015}(x)$ = $\Phi_{3\times 5\times 7\times 11\times 13}(x)$ has coefficients running from −22 to 22, $\Phi_{255255}(x)$ = $\Phi_{3\times 5\times 7\times 11\times 13\times 17}(x)$ , the smallest n with 6 different odd primes, has coefficients up to ±532.

Let A(n) denote the maximum absolute value of the coefficients of Φ_n. It is known that for any positive k, the number of n up to x with A(n) > n^k is at least c(k)⋅x for a positive c(k) depending on k and x sufficiently large. In the opposite direction, for any function ψ(n) tending to infinity with n we have A(n) bounded above by n^ψ(n) for almost all n.^[7]

Gauss's formula

Let n be odd, square-free, and greater than 3. Then^[8]^[9]

4\Phi_n(z) = A_n^2(z) - (-1)^{\frac{n-1}{2}}nz^2B_n^2(z)

where both A_n(z) and B_n(z) have integer coefficients, A_n(z) has degree φ(n)/2, and B_n(z) has degree φ(n)/2 − 2. Furthermore, A_n(z) is palindromic when its degree is even; if its degree is odd it is antipalindromic. Similarly, B_n(z) is palindromic unless n is composite and ≡ 3 (mod 4), in which case it is antipalindromic.

The first few cases are

\begin{align} 4\Phi_5(z) &=4(z^4+z^3+z^2+z+1)\\ &= (2z^2+z+2)^2 - 5z^2 \end{align}

\begin{align} 4\Phi_7(z) &=4(z^6+z^5+z^4+z^3+z^2+z+1)\\ &= (2z^3+z^2-z-2)^2+7z^2(z+1)^2 \end{align}

\begin{align} 4\Phi_{11}(z) &=4(z^{10}+z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z+1)\\ &= (2z^5+z^4-2z^3+2z^2-z-2)^2+11z^2(z^3+1)^2 \end{align}

Lucas's formula

Let n be odd, square-free and greater than 3. Then^[10]

\Phi_n(z) = U_n^2(z) - (-1)^{\frac{n-1}{2}}nzV_n^2(z)

where both U_n(z) and V_n(z) have integer coefficients, U_n(z) has degree φ(n)/2, and V_n(z) has degree φ(n)/2 − 1. This can also be written

\Phi_n((-1)^{\frac{n-1}{2}}z) = C_n^2(z) - nzD_n^2(z).

If n is even, square-free and greater than 2 (this forces n to be ≡ 2 (mod 4)),

\Phi_{n/2}(-z^2) = C_n^2(z) - nzD_n^2(z)

where both C_n(z) and D_n(z) have integer coefficients, C_n(z) has degree φ(n), and D_n(z) has degree φ(n) − 1. C_n(z) and D_n(z) are both palindromic.

The first few cases are:

\begin{align} \Phi_3(-z) &=z^2-z+1 \\ &= (z+1)^2 - 3z \end{align}

\begin{align} \Phi_5(z) &=z^4+z^3+z^2+z+1 \\ &= (z^2+3z+1)^2 - 5z(z+1)^2 \end{align}

\begin{align} \Phi_3(-z^2) &=z^4-z^2+1 \\ &= (z^2+3z+1)^2 - 6z(z+1)^2 \end{align}

Cyclotomic polynomials over Z_p

For any prime number p which does not divide n, the cyclotomic polynomial $\Phi_n$ is irreducible over Z_p if and only if p is a primitive root modulo n. That is, the p does not divide n, and its multiplicative order modulo n is $\varphi(n)$ (which is also the degree of $\Phi_n$ ).

Prime Cyclotomic numbers

The prime numbers of the form $\Phi_n(b)$ (with n, b integers, n > 2, b > 1) are listed in A206864, or all primes in A206942.

The list is about the smallest integer b > 1 which $\Phi_n(b)$ is a prime (see A085398), it is conjectured that such b exists for all positive integer n (See Bunyakovsky conjecture). (For that to allow b = 1, see A117544. In fact, b = 1 if and only if n is a prime or a prime power, so you can see this sequence for all positive integer n which is neither a prime nor a prime power. For n is a prime, see A066180).

The list is about all n ≤ 300 (The b-file of A117544 lists all n ≤ 1000, but it lists 1 if and only if n is a prime or prime power)

n	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11	+12	+13	+14	+15	+16	+17	+18	+19	+20
0+	3	2	2	2	2	2	2	2	2	2	5	2	2	2	2	2	2	6	2	4
20+	3	2	10	2	22	2	2	4	6	2	2	2	2	2	14	3	61	2	10	2
40+	14	2	15	25	11	2	5	5	2	6	30	11	24	7	7	2	5	7	19	3
60+	2	2	3	30	2	9	46	85	2	3	3	3	11	16	59	7	2	2	22	2
80+	21	61	41	7	2	2	8	5	2	2	11	4	2	6	44	4	12	2	63	20
100+	22	13	3	4	7	10	2	3	12	5	12	40	86	14	268	5	24	6	148	2
120+	43	2	12	6	127	2	2	102	2	3	7	3	2	5	33	56	13	8	11	4
140+	5	46	3	6	2	18	13	4	5	2	29	9	14	3	62	4	56	2	189	20
160+	3	93	30	12	2	49	44	18	24	2	22	14	60	2	63	17	47	16	304	35
180+	5	9	156	2	43	24	41	96	8	40	74	2	118	70	2	10	33	5	156	26
200+	41	2	294	16	11	5	127	2	103	25	46	41	206	6	167	88	39	12	105	15
220+	15	14	183	7	77	92	72	15	606	13	66	9	602	2	17	3	46	52	223	28
240+	115	19	209	61	67	11	15	5	27	25	37	23	69	2	3	120	52	17	69	28
260+	2	48	104	9	14	20	26	25	41	20	6	55	41	89	17	3	338	30	3	2
280+	217	34	13	69	112	14	3	5	315	65	15	196	136	22	44	2	56	16	219	4

For all positive integers n ≤ 1000, the largest three bs are 2706, 2061, and 2042, when n is 545, 601, and 943, and there are 17 values of n ≤ 1000 such that b > 1000.

In fact, if p is a prime, then $\Phi_p(b)$ is $\frac{b^p-1}{b-1}$ and a repunit number in base b, (111111...111111)_b, so the following is a list of the smallest b > 1 which $\Phi_p(b)$ is a prime. (see A066180)

The list is about the first 100 primes p. (The b-file of A066180 lists the first 200 primes p, up to 1223)

p	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71
min b	2	2	2	2	5	2	2	2	10	6	2	61	14	15	5	24	19	2	46	3
p	73	79	83	89	97	101	103	107	109	113	127	131	137	139	149	151	157	163	167	173
min b	11	22	41	2	12	22	3	2	12	86	2	7	13	11	5	29	56	30	44	60
p	179	181	191	193	197	199	211	223	227	229	233	239	241	251	257	263	269	271	277	281
min b	304	5	74	118	33	156	46	183	72	606	602	223	115	37	52	104	41	6	338	217
p	283	293	307	311	313	317	331	337	347	349	353	359	367	373	379	383	389	397	401	409
min b	13	136	220	162	35	10	218	19	26	39	12	22	67	120	195	48	54	463	38	41
p	419	421	431	433	439	443	449	457	461	463	467	479	487	491	499	503	509	521	523	541
min b	17	808	404	46	76	793	38	28	215	37	236	59	15	514	260	498	6	2	95	3

Applications

Using $\Phi_n$ , one can give an elementary proof for the infinitude of primes congruent to 1 modulo n,^[11] which is a special case of Dirichlet's theorem on arithmetic progressions.

Notes

↑ (sequence A013595 in OEIS)
↑ Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
↑ Schramm, Wolfgang (2015). "Eine alternative Produktdarstellung für die Kreisteilungspolynome". Elemente der Mathematik (Swiss Mathematical Society) 70 (4): 137–143. Retrieved 2015-10-10.
↑ Cox, David A. (2012), "Exercise 12", Galois Theory (2nd ed.), John Wiley & Sons, p. 237, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9 .
↑ Weisstein, Eric W. "Cyclotomic Polynomial". Retrieved 12 March 2014.
↑ Isaacs, Martin (2009). Algebra: A Graduate Course. AMS Bookstore. p. 310. ISBN 978-0-8218-4799-2.
↑ Meier (2008)
↑ Gauss, DA, Articles 356-357
↑ Riesel, pp. 315-316, p. 436
↑ Riesel, pp. 309-315, p. 443
↑ S. Shirali. Number Theory. Orient Blackswan, 2004. p. 67. ISBN 81-7371-454-1

References

Gauss's book Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

Gauss, Carl Friedrich (1986) [1801]. Disquisitiones Arithmeticae. Translated into English by Clarke, Arthur A. (2nd corr. ed.). New York: Springer. ISBN 0387962549.
Gauss, Carl Friedrich (1965) [1801]. Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory). Translated into German by Maser, H. (2nd ed.). New York: Chelsea. ISBN 0-8284-0191-8.
Lemmermeyer, Franz (2000). Reciprocity Laws: from Euler to Eisenstein. Berlin: Springer. doi:10.1007/978-3-662-12893-0. ISBN 978-3-642-08628-1.
Maier, Helmut (2008), "Anatomy of integers and cyclotomic polynomials", in De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian, Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13-17, 2006, CRM Proceedings and Lecture Notes 46, Providence, RI: American Mathematical Society, pp. 89–95, ISBN 978-0-8218-4406-9, Zbl 1186.11010
Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization (2nd ed.). Boston: Birkhäuser. ISBN 0-8176-3743-5.

External links

Weisstein, Eric W., "Cyclotomic polynomial", MathWorld.
Hazewinkel, Michiel, ed. (2001), "Cyclotomic polynomials", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Sloane's A013595 : Triangle of coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order)", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A013594 : Smallest order of cyclotomic polynomial containing n or −n as a coefficient", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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