Generalized Kummer theory and its applications

In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that ζ 6∈ k and ω ∈ k where ζ is a primitive n-th root of unity and ω = ζ + ζ−1. In particular, this result with ζ ∈ k implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.


Introduction
In this report we study the arithmetic of Rikuna's generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory.It is useful under the condition that ζ ∈ k and ω ∈ k where ζ is a primitive n-th root of unity and ω = ζ + ζ −1 .In particular, this result with ζ ∈ k implies the classical Kummer theory.We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.By an arithmetic argument we show that a certain cubic polynomial is not generic (cf.Corollary 3.6).
We first recall notion on the genericity of a polynomial (cf.Jensen-Ledet-Yui [3]).Let k be a field and G a finite group.The rational function field k(t 1 , t 2 , . . ., t m ) over k with m variables t 1 , t 2 , . . ., t m is denoted by k(t) where t = (t 1 , t 2 , . . ., t m ).For a polynomial F (X) ∈ K[X] over a field K let us denote by Spl K F (X) the minimal splitting field of F (X) over K.We say that a polynomial F (t, X) ∈ k(t)[X] is a k-regular G-polynomial or a regular polynomial over k for G if the field Spl k(t) F (t, X) is a Galois extension L of k(t) with two conditions Gal(L/k(t)) G and L ∩ k = k where k is an algebraic closure field of k.For example, if n is a positive integer greater than 2, then the Kummer polynomial X n − t ∈ Q(t)[X] is a regular polynomial for the cyclic group C n of order n not over Q but over Let n be an odd number greater than Proposition 1.1 (Rikuna [11]).The polynomial R n (t, X) is generic over the field k for the group C n .
Remark 1.2.When n is even and K does not contain ζ, the polynomial R n (t, X) is not generic over K for C n in general (cf.Komatsu [6]).For the case that n is even, Hashimoto and Rikuna [2] constructed a k-generic C n -polynomial with two parameters.
In a previous paper [6] we study the arithmetic of the polynomial R n (t, X).
Let k be a field whose characteristic is equal to 0 or prime to n.Let ζ be a primitive n-th root of unity in k and put ω , that is, . Let Γ K be the absolute Galois group Gal(K sep /K) of K where K sep is the separable closure field of K. Then we have a descent Kummer theory.
Proposition 1.3 (Ogawa [10], Komatsu [6]).There exists a group isomorphism We have a relation between the polynomial R n (t, X) and the algebraic group T as follows.For an s ∈ T (K) let L s be the field Spl K R n (s, X) and In particular, the field L s is equal to the fixed field (K sep ) Kerδ(s) of K sep by the subgroup Kerδ(s) of Γ K .

Ramifications and Artin symbols
In this section we recall some results in [6] and [7].Let l be an odd prime number and ζ a primitive l-th root of unity in Q.Let K be a finite algebraic number field containing Q(ω) where ω = ζ + ζ −1 .We assume that the extension K/Q(ω) is unramified at all the prime ideals of K above l.For an s ∈ K we denote by L s the minimal splitting field Spl K R l (s, X) of the polynomial R l (s, X) over the field K.For a prime ideal p of K let v p be a p-adic additive valuation which is normalized so that v p (K × ) = Z.
For a prime ideal l of K above l we define a set U K,l by For a prime ideal q of K with q l the set U K,q is defined to be Lemma 2.1 (Komatsu [6]).For an s ∈ K the conductor cond(L s /K) of the extension L s /K is equal to p p λp where Here c l (s) is equal to a positive integer We denote by U K the intersection ∩ p U K,p of the sets U K,p where p runs through all of the prime ideals of K.In general, one has that Let us assume that s The Galois group Gal(L s /K) is generated by an element σ such that Let p be a prime ideal of K which is unramified in the extension L s /K.We denote by F p the residue class field O K /p and by q the cardinal number F p of the finite field F p .Note that q ≡ 0 or ±1 (mod l) since K contains ω.We fix a prime ideal P of L s above p.Then there exists an element τ ∈ Gal(L s /K) such that v P (τ (α) − α q ) ≥ 1 for every algebraic integer α ∈ O Ls in L s .The element τ depends not on the choice of the prime ideal P but only on the prime ideal p.We call τ the Artin symbol of p in L s /K and denote it by Art p (L s /K).We put µ p (s) = v p (s 2 − ωs + 1).
When µ p (s) > 0 and µ p (s) ≡ 0 (mod l), the extension L s /K is totally ramified at p.
Theorem 2.3 does not deal with an exceptional case that µ p (s) > 0 and µ p (s) ≡ 0 (mod l), that is, µ p (s) = jl for a positive integer j ∈ Z.In the following we may reduce the exceptional case to the case µ p (s) ≤ 0. For a number s 0 ∈ K with v p (s − s 0 ) = j we put Lemma 2.4.We have L s = L s 1 and µ p (s 1 ) ≤ 0.
Proof.Corollary 1.5 shows that L s = L s 1 .Let p be a prime ideal of K(ζ) above p.Then one has that Proposition 2.5 (Komatsu [7]).We assume (l, K, p) = (3, Q, 3Z).For an s ∈ Q the decomposition of the prime ideal 3Z in the extension L s /Q is as follows: The number m at p-row in the table above means that s is a p-adic integer with s ≡ m (mod p).For example, if s ∈ Q satisfies that v 5 (s) ≥ 0 and s ≡ 1 (mod 5), then the ideal 5Z remains prime in L s /Q and the Artin symbol Art 5Z (L s /Q) is equal to σ 1 = σ.The symbol ∞ represents that v p (s) is negative, i.e., the image of s by the reduction map T (Q) → T (F p ), s → s (mod p) is equal to ∞.On the column of "ram.or bl.up", it holds that µ p (s) = v p (s 2 +s+1) ≥ 1.If µ p (s) is not divisible by 3, then p ramifies in L s /Q.When µ p (s) ≡ 0 (mod 3), one can blow-up s to a new s 1 ∈ Q such that L s = L s 1 and µ p (s 1 ) ≤ 0. In fact, for a number s 0 ∈ Q with v p (s − s 0 ) = µ p (s)/3 we put and µ p (s 1 ) ≤ 0. The decomposition type of p in L s /Q coincides with that in L s 1 /Q, which is determined completely by the data that s 1 belongs to the columns of "split" or "inert".In particular, p is unramified in L s /Q.The symbol − at the column of ram. or bl.-up is denoted for the fact that p ≡ 2 (mod 3) cannot ramify in any cyclic cubic fields due to class field theory.Indeed, it satisfies µ p (s) ≤ 0 provided p ≡ 2 (mod 3).The table for p = 3 is as follows.
For example, if s is a 3-adic integer with s ≡ 4 (mod 9), then 3Z ramifies in L s /Q.When v 3 (s) ≤ −2, the prime ideal 3Z splits completely in L s /Q.

Numerical examples of cubic polynomials
In this section we study the Artin symbols in the cyclic cubic fields obtained by some cubic polynomials.Let ζ be a primitive 3rd root of unity in Q.Let K be a field containing Q. Let f (X) be a cubic polynomial over which is the same as the starting one.Lecacheux [8] gave a cubic polynomial It is calculated that the discriminants discf i (t, X) of the polynomials f i (t, X) are Let c i (t) be rational functions in Q(t) such that Proof.The equations of the assertion follow from Lemma 3.1 and the algorithm for computing the invariants c = c i (t) of f i (t, X), respectively.Indeed, the square roots δ i (t) of the discriminants disc X f i (t, X) for the computations are For the polynomial f 0 ( c 1 (t), X) we have  Corollary 3.6.The polynomials f 1 (t, X) is not generic over Q for C 3 .Remark 3.7.By a geometric approach it is already shown that the polynomials f 1 (t, X), f 3 (t, X) and f 4 (t, X) are not generic for C 3 over any finite algebraic number fields (cf.[5]).Remark 3.8.There are symbols ∅ at 7-rows in the tables for f 0 ( c 1 (t), X) and f 0 ( c 3 (t), X), respectively.However, the case of Art 7 (M s /Q) = σ 2 occurs because of some blowing-up cases.
For a positive integer m ∈ Z let [m] be the multiplication map by m with respect to + T

Theorem 3 . 4 .Lemma 3 . 5 .
The family {Spl Q f 1 (s, X)|s ∈ Q} does not contain any cyclic cubic fields E which are unramified at two prime numbers 2 and 3with Art 2 (E/Q) = Art 3 (E/Q) = id.Let E 13 and E 19 be cyclic cubic fields with conductor 13 and 19, respectively.For i = 13 and 19 we have Art 2 (E i /Q) = Art 3 (E i /Q) = id, respectively.
The following table shows the Artin symbols Art p (L s /Q) for prime numbers p with 2 ≤ p ≤ 19 and p = 3.
The integer m at the p-row in the table above implies that s is a p-adic integer with s ≡ m (mod p).The symbol ∞ at the p-row means that v p (s) is negative.The notation m(p j ) represents that s is a p-adic integer with s ≡ m (mod p j ).For the polynomial f 0 ( c 3 (t), X) we have s /Q is ramified at 3. For the polynomial f 0 ( c 4 (t), X) we