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On the ring of integers of real cyclotomic fields.

1. Introduction. Let [[zeta].sub.n] be a primitive nth root of unity. It is well known that Z[[zeta].sub.n]] is the ring of integers of the nth cyclotomic field Q([[zeta].sub.n]]}. Generally this is proved by reducing the case of general n to the prime-power case (cf. [5]). On the other hand, Luneburg [4] directly proved the case of general n by showing that Z[[zeta].sub.n]] is a Dedekind domain.

It is also well known that Z[[[zeta].sub.n]] + [[[zeta].sup.-1.sub.n]]] is the ring of integers of the nth real cyclotomic field Q([[zeta].sub.n]] + [[zeta].sup.-1.sub.n]]}. This fact easily follows from the corresponding fact for Q([[zeta].sub.n]]} (cf. [5]). Another proof by use of the ramification groups is found in [3]. The purpose of this note is to give yet another proof of this fact, applying the method of [4] to Q([[zeta].sub.n]] + [[zeta].sup.-1.sub.n]]}. A key ingredient in the proof is the computation of the resultants of modified cyclotomic polynomials by the second named author in [8]. We also compute the discriminants of modified cyclotomic polynomials. We remark that analogous results have been obtained for cyclotomic function fields in [1].

2. Chebyshev polynomials and modified cyclotomic polynomials. We recall the definition of Chebyshev polynomials and modified cyclotomic polynomials, and quote some of their properties.

Definition 2.1. The Chebyshev polynomials [T.sub.n], [U.sub.n], V.sub.n] and [W.sub.n] of the first, second, third, and fourth kind, respectively, are characterized by

[T.sub.n] (cos [theta]} = cos n[theta], [U.sub.n](cos [theta]} = sin (n + 1)[theta]/sin [theta],

[V.sub.n] (cos[theta]) = cos (n + 1/2)[theta]/cos [theta]/2,

[W.sub.n] (cos[theta]) = sin (n + 1/2)[theta]/sin [theta]/2,

where n is a nonnegative integer. The normalized Chebyshev polynomials of the first and second kind are defined by [C.sub.n](x} = 2[T.sub.n](x/2}, [S.sub.n](x} = [U.sub.n](x/2}. We adopt Schur's notation [L.sub.n] = [S.sub.n-1]. For odd n we define [V.sub.n](x}= [V.sub.(n-1}]/2](x/2}, [W.sub.n](x} = [W.sub.(n-1}/2](x/2}.

Note that these polynomials all have integral coefficients.

Lemma 2.2.

(2.1} [C.sup.'sub.n] (x} = n[L.sub.n](x},

(2.2} [V.sup.'.sub.n](x} = n[W.sub.n](x) - [V.sub.n](x)/2(x + 2) (n : odd} and

(2.3} [W.sup.'.sub.n](x} = n[V.sub.n](x) - [W.sub.n](x)/2(x - 2) (n : odd}

We define the modified cyclotomic polynomials [[PSI].sub.n]. For n [greater than or equal to] 3 let [[PSI].sub.n] be the minimal polynomial of 2cos(2[pi]/n} over Q. Then [[PSI].sub.n](x) [member of] Z(x) and deg([[PSI].sub.n]} = [phi](n}/2. We do not define [[PSI].sub.1], [[PSI].sub.2] themselves, but instead we define their squares by

[[PSI].sub.1][(x}.sup.2] = x - 2, [[PSI].sub.2][(x}.sup.2] = x + 2.

Proposition 2.3 ([6, Proposition 2.4]).

(a) For n [greater than or equal to] 3, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) For odd n [greater than or equal to] 3, we have

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 2.4. Let n [??] 2 (mod 4). Let p be a prime and e,m positive integers such that n = [p.sup.e]m, p [??] m. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that the right side makes sense even if m = 1 since '(pe) is even by our assumption.

Proof. In the case where m [greater than or equal to] 3, this follows from [8, (3.4)].

In the case where m = 1, we are reduced to the case where n = p [greater than or equal to] 3 or n = 4, p = 2, by [8, (3.4)]. Suppose n = p [greater than or equal to] 3. By (2.5) and [7, Lemma 2.1 (ix)], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For n = 4, p = 2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (mod 2).

Since m [not equal to] 2, we complete the proof.

For a positive integer n let L(n) = p if n is a power of some prime p, and L(n) = 1 otherwise.

Lemma 2.5 ([8, Lemma 3.1]). Let n [greater than or equal to] 3.

(a) [[PSI].sub.n](2) = L(n).

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let res(f, g) denote the resultant of two polynomials f and g.

Theorem 2.6 ([8, Theorem 3.2]). Let 3 [less than or equal to] m < n.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. The discriminant of [[PSI].sub.n](x). We give an alternative proof of the following well known result. Let [DELTA](f) denote the discriminant of a polynomial f.

Proposition 3.1 ([2, Theorem 3.8]). Let n [greater than or equal to]

3. If n = [2.sup.e], e > 1, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If n = [p.sup.e] or n = 2[p.sup.e], p is an odd prime, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Otherwise,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Since all roots of [[PSI].sub.n] are real, [DELTA]([[[PSI].sub.n]) is positive. So we ignore the signs in the computation of [DELTA]([[PSI].sub.n]) throughout this proof.

Case n : odd If [[PSI].sub.n]([lambda]) = 0, by (2.3) and (2.5), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then it follows that

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [lambda] ranges over the roots of [[PSI].sub.n]. By (2.4) and Theorem 2.6 we have

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.5 (a) gives

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally we compute

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as follows: If n = [p.sup.e], then, by Theorem 2.6 we have

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then, by Theorem we have

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting (3.2) (3.6) into (3.1), the desired identity follows.

The argument is similar in the remaining cases, so we just provide some key identities:

Case n = 2 (mod 4)/ [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If n = 2[p.sup.e], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Case n [equivalent to] 0 (mod 4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the product being taken over all d such that d | n/4, d [not equal to] n/4, 4/4d is odd,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If n = [2.sup.e], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4. The ring of integers of Q([xi].sub.n + [[xi].sup.-1.sub.n]). We need the following lemma to prove Theorem 4.2. Let [F.sub.p] = Z/pZ.

Lemma 4.1 ([4, Hilfssatz 4]). Let [theta] be an algebraic integer, and f (x) the minimal polynomial of 9 over Q. Let P be a maximal ideal of Z[theta] and p the prime such that pZ = P [union] Z. Let [mu](x) be a monic polynomial over Z of least degree such that [mu]([theta]) [member of] P. Then, there exist polynomials g(x), h(x) [member of] Z[x] such that f = [mu]h + pg. Suppose gcd([mu],g, h) = 1 over [F.sub.p]. Then, the localization of Z[[theta]] at P is a discrete valuation ring.

The main result of this note is the following

Theorem 4.2. Let n [is greater than or equal to] 3. Then Z[[[xi].sub.n] + [[xi].sup.-1.sub.n]] is a Dedekind domain. Therefore Z[[[xi].sub.n] + [xi].sup.-1.sub.n]] is the ring of algebraic integers in the field Q([[[xi].sub.n] + [xi].sup.-1.sub.n]]).

Proof. We may assume n [not equal to] 2 (mod 4) since [[xi].sub.n] = [[xi].sub.n/2] if n = 2 (mod 4). Put [theta] = [[[xi].sub.n] + [xi].sup.-1.sub.n]], R = Z[[theta]]. We shall prove that R is a Dedekind domain by showing that the localization [R.sub.P] is a discrete valuation ring for each maximal ideal P [subset] R. Let p be the prime such that pZ = P [intersection] Z.

First we consider the case where p [??] n. By Proposition 3.1 we have p [??] [DELTA][[PSI].sub.n]), so that [PSI].sub.n](x) is separable over [F.sub.p]. If we apply Lemma 4.1 to f = [PSI].sub.n], then [mu](x) and h(x) have no common roots over the algebraic closure [bar.Fp], so [R.sub.P] is a discrete valuation ring.

Suppose p | n and write n = [p.sup.e]m, e [is greater than or equal to] 1, p [??] m.

By Lemma 2.4, there exists g(x) 2 Z[x] such that

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that [phi]([p.sup.e]) is even by our assumption n [??] 2 (mod 4).

Lemma 4.3. g([theta]) is a unit in R.

Proof. Suppose m [is greater than or equal to] 3. By Theorem 2.6 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the product being taken over the roots [lambda] of [[PSI].sub.n]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by (4.1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence [[PI].sub.[lambda]] = g([lambda])[+ or -]1, from which we conclude that g([theta]0) is a unit in R.

Suppose m = 1. By Lemma 2.5 we have

res([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence the claim follows similarly.

We return to the proof of Theorem 4.2. One could proceed as in [4], but here we give a shorter proof which was suggested to us by the referee. In the notation of Lemma 4.1, we take f, [mu], g such that f(x) = [[PSI].sub.n] (x) and g(x) is defined by (4.1). Since every root of [mu](x) mod P coincides with [theta] mod P for some choice of primitive nth root of unity [[xi].sub.n], we see, by Lemma 4.3, that [mu](x) and g(x) have no common roots over [bar.Fp]. Hence [R.sub.P] is a discrete valuation ring by Lemma 4.1. This completes the proof.

Acknowledgment. The authors would like to thank an anonymous referee for pointing out the simplification of the proof of Theorem 4.2.

References

[1] S. Jeong, Resultants of cyclotomic polynomials over [F.sub.q][T] and applications, Commun. Korean Math. Soc. 28 (2013), no. 1, 25 38.

[2] D. H. Lehmer, An extended theory of Lucas' functions, Ann. of Math. (2) 31 (1930), no. 3, 419 448.

[3] J. J. Liang, On the integral basis of the maximal real subfield of a cyclotomic field, J. Reine Angew. Math. 286/287 (1976), 223 226.

[4] H. Luneburg, Resultanten von Kreisteilungspolynomen, Arch. Math. (Basel) 42 (1984), no. 2, 139 144.

[5] L. C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, 83, Springer, New York, 1997.

[6] M. Yamagishi, A note on Chebyshev polynomials, cyclotomic polynomials and twin primes, J. Number Theory 133 (2013), no. 7, 2455 2463.

[7] M. Yamagishi, Periodic harmonic functions on lattices and Chebyshev polynomials, Linear Algebra Appl. 476 (2015), 1-15.

[8] M. Yamagishi, Resultants of Chebyshev polynomials: the first, second, third, and fourth kinds, Canad. Math. Bull. 58 (2015), no. 2, 423 431.

By Koji YAMAGATA * and Masakazu YAMAGISHI **

(Communicated by Shigefumi Mori, M.J.A., May 12, 2016)

2010 Mathematics Subject Classification. Primary 11E09; Secondary 11R18.

* Field of Mathematics and Mathematical Science, Department of Computer Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi 466-8555, Japan.

** Department of Mathematics, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi 466-8555, Japan.
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Author:Yamagata, Koji; Yamagishi, Masakazu
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:Jun 1, 2016
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