The "Bible" of Algebraic Numbers:
In his famous "Zahlbericht" [1] HILBERT (1862 - 1943) states a series of theorems (nrs. 90 to 94) on relatively cyclic fields that forms the basis for the impressive building of classfield theory, which has been braught to perfection by Weber, Takagi, Artin, Furtwängler, and Hasse.
Three of these theorems and certain extensions are essential for my classification of cubic fields according to their ambiguous principal ideals and shall be expressed here in different formulations.
We assume that K | k is a cyclic relative extension of algebraic number fields with relative Galois group G = Gal(K | k) generated by a substitution S, i. e., G = < S >.
Theorem 90:
As a theorem on numbers with relative norm 1: Any number A in K with relative norm N_{k}( A ) = 1 with respect to k is the symbolic ( 1 - S )th power of an algebraic integer B in K, A = B^{1 - S}.
As a theorem of homological algebra: K^{x} * ker( N_{k} ) / ( K^{x} )^{1 - S} = H^{1}( G, K^{x} ) = 1. The 1st cohomology group of G in K^{x} is trivial. (* denotes the intersection)
Generalization: Now let K | k be a GALOIS relative extension of algebraic number fields.
Theorem:
Any 1-cochain of G in K^{x}, ( A_{S} )_{S in G} in ( K^{x} )^{G}, which is a crossed homomorphism, i. e., A_{ST} = A_{S} A_{T}^{S} for all S,T in G, can be represented as a 1-coboundary of G in K^{x}, ( A_{S} )_{S in G} = ( B^{1 - S} )_{S in G} with an algebraic integer B of K.
Assumptions: From now on let the relative degree [ K : k ] be a prime number p. In the case p = 2, suppose that the number of real conjugate fields of K is twice the number of real conjugate fields of k.
Theorem 92:
As a theorem on the existence of a unit with relative norm 1, that cannot be represented as a quotient of two relatively conjugate units: In K there exists a unit H with relative norm N_{k}( H ) = 1 with respect to k, which is not the symbolic ( 1 - S )th power of a unit in K.
As a theorem of homological algebra: U_{K} * ker( N_{k} ) / U_{K}^{1 - S} = H^{1}( G, U_{K} ) < > 1. The 1st cohomology group of G in U_{K} is not trivial. (* denotes the intersection and U_{K} * ker( N_{k} ) will be abbreviated as E_{K|k})
Theorem 94:
Fundamental theorem on relatively cyclic fields with the relative different 1. Let the relative different of K with respect to k be D_{k} = (1). Then we have the following facts: There exists an ideal j in k, which is not a principal ideal in k, but becomes principal in K: j = ( A ). The p-th power of this ideal is a principal ideal already in k, j^{p} = ( N_{k}( A ) ), and therefore its class has order p, [ j ]^{p} = 1. Another formulation: neither the principalization kernel P_{K} * I_{k} / P_{k} of K | k nor the p-elementary class group C_{k,p} of k are trivial, and K is an (unramified) HILBERT classfield of k. (* denotes the intersection)
Application to Cubic Fields:
The normal field N of a non-Galois cubic field K is cyclic of prime degree [ N : k ] = 3 over its quadratic subfield k. A fundamental unit e in the subgroup E_{0} of E_{N|k} generated by all subfield units of N ("old units" in the terminology of A. Scholz) with positive norm can be represented as a quotient of relatively conjugates of an algebraic integer in N, according to Hilbert's Theorem 90. For the shape of this algebraic integer of N we have several possibilities (cfr. [2] ):
e = A^{1-S} | e = B^{1-S} | e = E^{1-S} | e = H^{1-S} | e = Z^{1-S} |
(A) = a | (B) = L^{1-T} | E in E_{N|k} | N_{k}(H) = eta | N_{k}(Z) = zeta |
In the first two cases, the fundamental unit e of the cubic field K can be represented as a quotient e = X^{1 - S} of relatively conjugates in the normal field N only by means of a non-unit X.
1. e = A^{1 - S} with a non-unit A of N, whose principal ideal (A) = a is an extension ideal of an ideal a of k but not of a principal ideal of k (more detailed: AO_{N} = aO_{N} with the maximal order O_{N} of N). According to a definition of Moriya, e is an element of the group E_{1} of relative units of the 1st kind of N | k and the principal ideal (A) = a lies in the capitulation kernel P_{N} * I_{k} / P_{k} of N | k. (* denotes the intersection)
2. e = B^{1 - S} with a non-unit B of N, whose principal ideal (B) = L^{1 - T} is composed in a characteristic manner of the prime ideals L_{1},L_{1}^{T},...,L_{s},L_{s}^{T} of N lying over the finitely many prime ideals l_{1},...,l_{s} of K that split in N | K and divide the relative different of N | k: L is a certain product of these overlying prime ideals and of their squares. Hence the principal ideal (B) = L^{1 - T} belongs to the group D_{N|k}^{-}I_{k} / I_{k} of relatively ambiguous principal ideals of N | k and thus it lies in the kernel of the norm of N | K. Here we define D_{N|k}^{-} = P_{N} * < L_{1}^{1 - T},...,L_{s}^{1 - T} >. (* denotes the intersection)
In the last three cases, the fundamental unit e of the cubic field K can be represented as a quotient e = Y^{1 - S} of relatively conjugates in the normal field N with the aid of a unit Y.
3. e = E^{1 - S} with a relative unit E of N | k. Then we have e = a^{3} / N(a) with an absolutely ambiguous principal ideal (a) of K in the group P_{K}^{G} / P_{Q}, where G = < S >. The relative unit E itself could have a representation of the form E = b^{1 - S} with the non-unit b = 1 / S^{2}(a), in this case.
4. e = H^{1 - S} with a unit H, that has the fundamental unit of k as its norm N(H) = eta, if k is real quadratic and thus K is a totally real cubic field. Therefore neither can e contribute to generating E_{N|k} modulo U_{N}^{1 - S}, since e lies in U_{N}^{1 - S}, nor can H, since H is not a relative unit.
5. e = Z^{1 - S} with a unit Z with norm N(Z) = zeta, the primitive 3rd root of unity in the cyclotomic field k = Q(zeta), in the case of a pure cubic field K. As before, neither can e contribute to generating E_{N|k} modulo U_{N}^{1 - S}, since e belongs to U_{N}^{1 - S}, nor can Z, since Z is not a relative unit.
Bibliography:
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