# Pure Quintic Fields 1

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 Pure Quintic Fields 1 Previous Page § 3. Conductor and discriminants. Let R = q1 … qs be the squarefree product of all prime divisors of the radicand D of the pure quintic field L = Q(D1/5). Independently of the exponents e1,…,es, the conductor f of the cyclic quintic relative extension N|k is given by f4 = 52 R4 if not D ≡ ±1,±7 (mod 25) (field of the first kind), f4 = R4 if D ≡ ±1,±7 (mod 25) (field of the second kind) [Thm. 1, p. 103, Ma1]. Examples 3.1. The smallest radicands D of pure quintic fields L = Q(D1/5) with given kind and increasing number of prime divisors are D = 2, 6 = 2*3, 42 = 2*3*7, for fields of the first kind with (D,5) = 1, D = 5, 10 = 2*5, 30 = 2*3*5 , for fields of the first kind with 5|D, D = 7, 18 = 2*32, 126 = 2*7*32, for fields of the second kind. It is well known that the cyclotomic discriminant takes the value d(k) = +53 = 125, and Hilbert's Theorem 39 on discriminants of composite fields shows the following Theorem 3.1. The metacyclic discriminant is given by d(N) = d(k)5 f16 = = 523 R16 for a field of the first kind, = 515 R16 for a field of the second kind, the intermediate discriminant is given by d(M) = 5 d(k)2 f8 = = 511 R8 for a field of the first kind, = 57 R8 for a field of the second kind, and the pure quintic discriminant is given by d(L) = d(k) f4 = = 55 R4 for a field of the first kind, = 53 R4 for a field of the second kind. § 4. Herbrand quotient of units of N|k. In section 2, we have seen that the unit index (UN:U0) = 5e admits a coarse classification of pure quintic fields according to the seven possible values of the exponent 0 ≤ e ≤ 6. However, the Galois cohomology of the unit group UN with respect to the cyclic quintic Kummer extension N|k provides additional structural information. The Herbrand quotient of UN under the action of Gal(N|k) = < σ > is defined by #H0(Gal(N|k),UN) / #H-1(Gal(N|k),UN) = = #[ker(Δ)/im(Norm)] / #[ker(Norm)/im(Δ)] = = # [Uk/NormN|kUN] / # [EN|k/UN1 - σ], where Δ : E → E1 - σ and Norm : E → E1 + σ+ … + σ4 are endomorphisms of UN and EN|k consists of the units E in UN with relative norm NormN|k(E) = 1. According to Takagi (1920), Hasse (1927) and Herbrand (1932), the Herbrand quotient of the Gal(N|k)-module UN has the value #H0(Gal(N|k),UN) / #H-1(Gal(N|k),UN) = 1 / [N : k], since Archimedean places do not yield a contribution. Next Page