A polynomial related to the cyclotomic polynomial

(1)

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S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

L. C ARLITZ

A polynomial related to the cyclotomic polynomial

Rendiconti del Seminario Matematico della Università di Padova, tome 47 (1972), p. 57-63

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TO THE CYCLOTOMIC POLYNOMIAL

L. CARLITZ

*)

1. Let

denote the

cyclotomic polynomial.

^In^a^recentpaper

[ 1 ] Apostol

^has:

determined the resultant

R(Fm , Fn)

of the

cyclotomic polynomials ^Fm(x),.

[3].

For m, put

where ’(1., runs

through

^the

primitive

^m-th^roots^of

unity

^and

5

^runs

through

^the

primitive

^n-th^roots^of

unity. Clearly

Also it follows at once

from (1.2)

^that

*) Indirizzo dell’A.: Dept. ^ofMathematics - Duke University - Durham,.

North Carolina 27706, ^U.S.A.

Supported in part by NSF grant GP-17031.

(3)

58

where

and

In

(1.5),

^asⁱⁿ

(1.2), a

^runs

through

^the

primitive

^m-th^roots^of

unity

while

’~

^runs

through

^the

primitive

^n-th^roots^of

unity.

Let k denote the

greatest

^common^divisorof rn and n such that

We _maycall k the

unitary greatest

^commondivisor of m and n and write

(We

^remark^that^Eckford^Cohen

[2]

^has^defined^asimilar function that is denoted

by ^{(m, n)*).}

^Weshall show

that,

^for

k= 1,

where

The

general

^case

^{(k ? 1 )}

^is

given

ⁱⁿ^Theorem¹^below.

Apostol’s

result is ^an_easy

corollary

^ofthe theorem. Indeed a

slightly

^more

general

^result^is

given

ⁱⁿ

(3.12).

2. We shall

require

^several^lemmas.

LEMMA 1. The number

of

^solutions

of

where p is ^a

prime

^ande >

1,

^is

equal

^to

(4)

LEMMA 2. Let

f(a, n)

denote the number

of

^solutions

of

Then, for (m, n) = 1,

The

proof

ôf^these^two^{lemmas is}âlmostîmmediate.

LEMMA 3. Let .a,

~3 independently

^run

through

^the

primitive

^k-th

roots

of unity.

^Let^r^denote^an

arbitrary

^divisor

of

^k.^Then^the

primitive

r-th roots

of unity

^occur

k)

^timesamong the

products

^where

PROOF. Let E denote a fixed

primitive

^k-th^root^of

unity

and put

a==~~ ~==~, where x,

y~ ^run

through

reduced residue systems

(mod k) .

Let k=rs. Then is a

primitive

^r-th^root^of

unity

^if^and

only

^if

where

(a, r) =1.

For fixed _{a, s,}the number of solutions of this con-

gruence is

f (as, k)

^as^defined^{in Lemma}^2.^Now

apply

^Lemmas¹^’and

2

and,(2.5)

^follows^at^once.

LEMMA 4. Let

(m, n)* =1,

^where

(m, n)*

^is

defined by ( 1.7)

^and

( 1.8).

^Let^a^run

through

^the

primitive

^m-th^roots

of unity

^and

5 through

the

primitive

^n-th^roots

of unity.

^Then

y=a5

^runs

through

^the

primitive

M-th roots

of unity g(d) times,

where

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60

It suffices ^to_provethis when

where _pis

prime.

^{Then if}ôcîsâ

primitive pe-th

^root^of

unity and

is a

primitive pf-th

^root,^itis clear that

-t=a5

^is^a

primitive pe-th

^root.

Moreover each _ywill occur

exactly

^times.

3. Given _m,

n >_ 1,

^define

k= (m, n)* by

^means^of

(1.7)

^and

(1.8)

and put

so that

Then

by ^(1.5)

^and

^{( 3 .1 )}

^we^have

where

a(k), 5(k) independently

^run

through

^the

primitive

^k-th^roots^of

unity

^while

a(m’), 0(n’)

^run

through

^the

primitive

m’-th and n’-th roots,

respectively. Applying

^Lemmas³^and

(4)

^weget

where in the inner

product ~y{r)

^runs

through

^the

primitive

^r-th^roots

of

unity

^and

~a{M)

^runs

through

^the

primitive

M-th roots,

Since

(r, M) =1,

^runs

through

^the

primitive

^rM-th^roots

of

unity.

^Hence

(3.3)

^becomes

(6)

Making

^use^ofthe formula

we

get

where

Put

It follows from

(2.5)

^and

(3.6)

^that

Moreover,

when k* ~

s,

^weget

where s= II

pl.

We _maynow state the

following

THEOREM 1. Let k be the greatest ^commondivisor

of

^m^andⁿ

such that

and put

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62

Then

where the

product

^isrestricted to those s that are divisible

by

^{k* and}

e(s,

u,

k)

^is^evaluated

by ^(3.9).

^In

particular, if k=1,

^then

where

M=![m, n], d=(m, n).

It is

easily

verified that

Hence

(3. 10) implies

provided M = pa.

Thus if m > n > 1 and then

in

agreement

^with

Apostol [ 1 ] .

It _maybe of interest to mention that

if ~

^denotes^a^k*-th^root^of

unity (not necessarily primitive)

^then^we^have

where

again m = n pa.

REFERENCES

[1] APOSTOL, ^{T. M.:}Resultants of cyclotomic polynomials, Proceedings ^{of the}

American Mathematical Society, ^vol.²⁴(1970), 457-462.

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[2] ECKFORD, COHEN: Arithmetical functions associated with the unitary ^divisors of ^aninteger, Mathematische Zeitschrift, ^vol.⁷⁴(1960), ^66-80.

[3] DIEDERICHSEN, ^F.^E.: Über die Ausreduktion ganzzahliger Gruppendarstel- lungen bei arithmetischer Äquivalenz, Abh. Math. Sem. Hansischen Univ., vol. 13 (1964), ^35,7-412.

Manoscritto pervenuto ⁱⁿredazione il 24 settembre 1971.

A polynomial related to the cyclotomic polynomial

R ENDICONTI

del

S EMINARIO M ATEMATICO

della

U NIVERSITÀ DI P ADOVA

L. C ARLITZ

A polynomial related to the cyclotomic polynomial

Rendiconti del Seminario Matematico della Università di Padova, tome 47 (1972), p. 57-63

*)

cyclotomic polynomial.

[ 1 ] Apostol

R(Fm , Fn)

cyclotomic polynomials Fm(x),.

[3].

through

primitive

unity

5

through

primitive

unity. Clearly

from (1.2)

(1.5),

(1.2), a

through

primitive

unity

’~

through

primitive

unity.

greatest

unitary greatest

(We

[2]

by (m, n)*).

that,

k= 1,

general

(k ? 1 )

given

Apostol’s

corollary

slightly

general

given

(3.12).

require

of

of

prime

1,

equal

f(a, n)

of

of

Then, for (m, n) = 1,

proof

~3 independently

through

primitive

of unity.

arbitrary

of

primitive

of unity

k)

products

primitive

unity

a==~~ ~==~, where x,

through

(mod k) .

primitive

unity

only

(a, r) =1.

f (as, k)

apply

R ^ENDICONTI

S ^EMINARIO M ^ATEMATICO

U NIVERSITÀ DI P ^ADOVA

cyclotomic polynomials ^Fm(x),.

by ^{(m, n)*).}

^{(k ? 1 )}

n >_ 1,

by ^(1.5)

^{( 3 .1 )}

by ^(3.9).