On the determination of number fields by L-functions

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Florin NICOLAE

On the determination of number fields by L-functions Tome 28, no_{2 (2016), p. 431-433.}

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Journal de Théorie des Nombres de Bordeaux 28 (2016), 431–433

On the determination of number fields by

L-functions

par Florin NICOLAE

Résumé. Nous montrons qu’une extension galoisienne du corps des nombres rationnels est déterminée par toute fonction L d’Artin corespondant à un caractère du groupe de Galois qui contient tous les caractères irréductibles.

Abstract. We prove that a Galois number field is determined by any Artin L-function corresponding to a character of the Galois group which contains all irreducible characters.

Let K/Q be a finite Galois extension with the Galois group G, χ1, . . . , χr

the irreducible characters of G with the dimensions d1 := χ1(1), . . . , dr :=

χr(1), and L(s, χ1, K/Q), . . . , L(s, χr, K/Q) the corresponding Artin

L-functions. The Dedekind zeta function ζK(s) is the Artin L-function

corre-sponding to the regular character Reg_G=Pr

j=1djχj:

ζK(s) = L(s, RegG, K/Q)

= L(s, χ₁_{, K/Q)}d1_{· . . . · L(s, χ}

r, K/Q)dr.

It is known that ζ_K_{(s) determines the field K: if L/Q is a finite normal} extension of Q such that

ζL(s) = ζK(s)

then

L = K.

Since d_j > 0, the regular character Reg_G contains all irreducible characters

χj, j = 1, . . . , r. Starting from this observation, we prove that the field K

is determined by any Artin L-function corresponding to a character of the Galois group G which contains all irreducible characters.

Manuscrit reçu le 26 mars 2014, révisé le 16 juillet 2014, accepté le 18 septembre 2014.

Mathematics Subject Classification. 11R42. Mots-clefs. Galois number field, Artin L-function.

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432 _{Florin Nicolae}

Theorem 1. Let K/Q, L/Q be finite Galois extensions, G the Galois group

of K/Q, χ1, . . . , χr the irreducible characters of G, k1 > 0, . . . , kr> 0

inte-gers, χ =Pr

j=1kjχj, H the Galois group of L/Q, ϕ1, . . . , ϕtthe irreducible

characters of H, l1 > 0, . . . , lt> 0 integers, ϕ =Ptj=1ljϕj. If

L(s, χ, K/Q) = L(s, ϕ, L/Q) then

K = L.

Proof. Let M/Q be a finite Galois extension which contains both K and L. For j = 1, . . . , r let

˜

χj : Gal(M/Q) → C, ˜χj(σ) := χj(σGal(M/K)),

where we identify G with the factor group Gal(M/Q)/Gal(M/K). Then ˜χj

is an irreducible character of Gal(M/Q) and (1) Gal(M/K) = ∩r_j=1Ker ( ˜χj),

where Ker ( ˜χj) is the kernel of the character ˜χj. For j = 1, . . . , t let

˜

ϕj : Gal(M/Q) → C, ˜ϕj(σ) := ϕj(σGal(M/L)),

where we identify H with the factor group Gal(M/Q)/Gal(M/L). Then ˜ϕj

is an irreducible character of Gal(M/Q) and (2) Gal(M/L) = ∩t_j=1Ker ( ˜ϕj).

It is a fundamental property of Artin L-functions that

L(s, χj, K/Q) = L(s, ˜χj, M/Q), j = 1, . . . , r and that L(s, ϕj, L/Q) = L(s, ˜ϕj, M/Q), j = 1, . . . , t hence L(s, χ, K/Q) = r Y j=1 L(s, χj, K/Q)kj = r Y j=1 L(s, ˜χj, M/Q)kj, and L(s, ϕ, L/Q) = t Y j=1 L(s, ϕj, L/Q)lj = t Y j=1 L(s, ˜ϕj, M/Q)lj, therefore (3) r Y j=1 L(s, ˜χj, M/Q)kj = t Y j=1 L(s, ˜ϕj, L/Q)lj.

Artin proved that the L-functions corresponding to irreducible characters are multiplicatively independent ([1], Satz 5, P. 106). Thus, from the mul-tiplicative relation (3) we deduce that

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Determination of number fields 433 Together with (1) and (2), this implies that

Gal(M/K) = Gal(M/L), so

K = L.

References

[1] E. Artin, “Über eine neue Art vonL-Reihen”, Abh. Math. Sem. Univ. Hamburg 3 (1924), no. 1, p. 89-108.

Florin Nicolae

Technische Universität Berlin Institut für Mathematik Strasse des 17. Juni 136 D-10623 Berlin, Germany

and

Institute of Mathematics of the Romanian Academy P.O.BOX 1-764

RO-014700 Bucharest, Romania

E-mail: Florin.Nicolae@imar.ro URL: http://www.imar.ro/