Kummer extensions of number fields (the case of rank 2)

(1)

KUMMER EXTENSIONS OF NUMBER FIELDS (THE CASE OF RANK 2)

ANTONELLA PERUCCA

ABSTRACT. LetKbe a number field, and let`be a prime number. Fix some elementsα1,α2

ofK^×which generate a torsion-free subgroup ofK^×of rank2. Letn1,n2,mbe positive integers withm>max(n1, n2). We show that there exist parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extensionK(ζ`^m, ^`n1√

α1, ^`n2√ α2) overK(ζ`^m)for alln1,n2,m. The parameters appearing in the formulas are explicitly computable.

1. INTRODUCTION

Let K be a number field, and let `be a prime number. Letα₁, . . . , α_r be elements ofK^× which generate a torsion-free subgroup of K^× of rank r. Ifn, m are positive integers with m>n(and ifζ_`^mdenotes a primitive`^m-th root of unity) then we are interested in the degree of the Kummer extension

K(ζ`^m, ^`n√

α1, . . . , ^`n√ αr)

overK(ζ_`^m) for allnandm. Together with Debry [1] and with Sgobba and Tronto [5] the author proved that there are finitely many divisibility parameters such that the above Kummer degree can be expressed with parametric formulas for allnandm. There is only a small case distinction if` = 2andζ₄ ∈/ K, and the parameters appearing in the formulas are explicitly computable. A natural generalization is the following:

Question 1. Letn1, . . . , nr,mbe positive integers withm>maxi(ni). Consider the Kummer extension

(1) K(ζ_`^m, ^`n1√

α₁, . . . , ^`nr√ α_r)

overK(ζ_`^m)for alln₁, . . . , n_r, m. Can we express the Kummer degree for alln₁, . . . , n_r, m with parametric formulas involving only a finite case distinction and finitely many computable parameters?

This question would be trivial ifr = 1or if we fixn1, . . . , nr, m, so if we consider only one Kummer degree (because we can reduce to the above-mentioned results).

The question is also trivial if the elementsα1, . . . , αr are strongly`-independent in the sense of [1, Definitions 10 and 5], and we have`6= 2orζ₄∈K, because in this case we simply have

(2) [K(ζ`^m, ^`n1√

α1, . . . , ^`nr√

αr) :K(ζ`^m)] =`^P^rⁱ⁼¹ⁿⁱ,

2010Mathematics Subject Classification. Primary: 11Y40; Secondary: 11R20, 11R21.

Key words and phrases. Number field, Kummer theory, Kummer extension, degree.

1

(2)

see Remark 11. However, the question in general does not seem to be easy. In this short note we answer Question 1 in the affirmative for rank2:

Theorem 2. LetK be a number field, and let`be a prime number. Fix some elementsα1, α2ofK^×which generate a torsion-free subgroup ofK^×of rank2. Letn1,n2,mbe positive integers withm>max(n₁, n₂). We can express the degree of the Kummer extension

(3) K(ζ`^m, ^`n1√

α1, ^`n2√

α2)/K(ζ`^m)

for alln1, n2, mwith parametric formulas involving only a finite case distinction and finitely many computable parameters.

The case distinction and the proof of the theorem can be found in Section 4. We have a more precise result ifζ`∈/K, see Section 4.3.

It is possible that the method described in this article can be used to generalize Theorem 2, however it can be expected that the case of rank 3 (or higher) requires a more involved case distinction. Notice that if finding the parametric formulas proves to be unfeasible, one could still try to argument that such formulas must exist. We leave this as a direction of future research.

For recent works about related problems see the PhD thesis of Palenstijn [2] and the following articles of the author with Sgobba and Tronto [4, 6, 7].

Acknowledgments. The author would like to thank Pietro Sgobba for a careful reading of the manuscript.

2. PRELIMINARIES

2.1. Notation. From now onKis a number field and`is a fixed prime number. Writeµ_K,`^∞ for the group of roots of unity inK^×having order a power of`. Callωthe greatest nonnegative integer such thatK^× contains a primitive`^ω-th root of unity. We writev_`to mean the`-adic valuation; for a root of unity we writeord`to mean the`-adic valuation of its order. Ifn>1, we writeζ_`ⁿto mean a primitive root of unity of order`ⁿ.

We callα ∈ K^×strongly indivisibleifαis not an`-th power inK^×times a root of unity in µ_K,`^∞. Notice that ifα is strongly indivisible, then it is also not an`-th power in K^× times a root of unity inK^×(this is immediate to verify). Also notice that if ζ_` ∈/ K, then strongly indivisible just means not being an`-th power inK^×.

Ifα₁, . . . , α_rare elements ofK^×, then we say that they arestrongly independentif an element ofK^× of the formα^e₁¹· · ·α_r^e^r (wheree1, . . . , er are integers) is strongly indivisible unless` divides all exponentse₁toe_r. For a single element the two notions of strongly indivisible and strongly independent are the same.

2.2. Divisibility parameters. Ifα∈K^×is not a root of unity, we can express it as α=A^`^dξ

(3)

for some nonnegative integerd, for some elementA∈ K^×which is strongly indivisible, and for some root of unity ξ inµ_K,`^∞ having order `^h. We call dthe d-parameterandh theh- parameter: these two nonnegative integers depend onαandK(the prime`is fixed). It is not difficult to prove that the d-parameter is uniquely determined, which means that it does not depend on the chosen decomposition (see [3, Lemma 8]). Moreover, the decomposition ofα can be chosen in such a way that eitherh = 0orh > max(0, ω−d): with these additional constraints,his uniquely determined (see [3, Lemma 8]).

IfGis a finitely generated and torsion-free subgroup ofK^×of positive rankr, then we define two sets ofrparameters which we call the d-parameters and the h-parameters respectively. Let G=hα₁, . . . , αriand for eachi= 1, . . . , rlet

αi =A^`_i^diξi

for some nonnegative integerd_i, for some elementA_i∈K^×which is strongly indivisible, and for some root of unityξi inµK,`^∞having order`^hⁱ. We callα1, . . . , αragood basisforGif A₁, . . . , A_r are strongly independent. This is equivalent to the fact thatPr

i=1d_i (the sum of the d-parameters) is maximal among the possible bases ofG(see [1, Section 3]). The group Galways admits a good basis by [1, Theorem 14]. Every good basis has the same multiset of d-parameters (see [1, Corollary 16]), while the multiset of h-parameters is not unique. The h- parameters would become unique if we reorder the good basis ofGsuch that the d-parameters are ordered non-decreasingly and we require the following conditions:

(1) For every16i6rwe havehi= 0orhi > ω−di.

(2) If16i < j 6randhi, hj >0hold, then we havehi > hj anddi+hi< dj+hj. (3) If16i < j 6randdi =djhold, thenhj = 0.

The non-uniqueness of the h-parameters for a good basis is discussed in [1, Appendix]: to gain flexibility we prefer not to impose the above conditions on the h-parameters. We call the above parameters the d-parameters and the h-parameters forG, keeping in mind that the h-parameters are not uniquely determined.

2.3. Constructing a good basis. LetGbe a finitely generated and torsion-free subgroup of K^×of positive rankr. Letα1, . . . , αrbe a basis ofGwith d-parametersd1, . . . , dr.

Remark 3. As shown in[1, Proof of Theorem 14], if the given basis ofGis not a good basis, then there is a base change altering only one of the elementsαi’s and such that the sum of the d-parameters for the new basis is greater thanPr

i=1d_i (this simply amounts to the fact that the new generator has a greater d-parameter with respect to the replaced generator). We can prove that we have a good basis by showing that a base change as above does not exist.

A good basis is explicitly computable, as explained in [1, Section 6.1].

3. COMPUTING DIVISIBILITY PARAMETERS

LetKbe a number field and`a fixed prime number.

(4)

Remark 4. LetA be an element ofK^×which is strongly indivisible. Then by definition for every non-zero integerXand for everyζ ∈µ_K,`^∞the elementA^Xζhasv_`(X)as d-parameter andord`ζ as h-parameter.

Lemma 5. Let A₁, . . . , A_n be elements of K^× which are strongly independent. Then for all non-zero integers X1, . . . , Xn and for every ζ ∈ µK,`^∞ the elementζQn

i=1A^X_i ⁱ has d- parametermin_i(v_`(X_i))and h-parameterord_`ζ.

Proof. Callxi =v_`(Xi)andm= mini(xi). The element C:=

n

Y

i=1

A^X_i ⁱ^/`^m

is the product of powers of strongly independent elements whose exponents are not all divisible by`and hence it is strongly indivisible (by definition of strongly independent). So we can write

ζ

n

Y

i=1

A^X_i ⁱ =C^`^mζ

and from the latter decomposition we can easily read off the divisibility parameters.

Lemma 6. LetA₁, . . . , A_nbe elements ofK^×which are strongly independent, and letζ₁, . . . , ζ_n be roots of unity inµ_K,`^∞. Then for all non-zero integersX₁, . . . , X_nthe elements

A^X₁¹ζ₁, . . . , A^X_nⁿζ_n

form a good basis for the group that they generate. The (d, h)-parameters for this group are (v_`(Xi),ord_`ζi)fori= 1, . . . , n.

Proof. CallGthe grouphA^X₁¹ζ₁, . . . , A_n^Xⁿζ_ni, and writex_i =v_`(X_i). The(d, h)-parameters for the given basis are(xi,ord`ζi)by Remark 4, so we are left to prove that we have a good basis. If not, then by Remark 3 it is possible to replace w.l.o.g.A^X₁¹ζ1by

C :=

n

Y

i=1

A^X_i ⁱ^Eⁱ·

n

Y

i=1

ζ_i^Eⁱ

for some integers E_i not all zero such that G = hC, A^X₂²ζ₂, . . . , A^X_nⁿζ_ni and such that the d-parameter ofCis greater thanx1. The first condition forcesv`(E1) = 0becauseA^X₁¹ times a root of unity belongs toG, so the d-parameter ofCis at mostv_`(X₁E₁) =x₁by Lemma 5,

contradiction.

Lemma 7. LetGbe a torsion-free subgroup ofK^×of rankr. LetE be an integer coprime to

`, and letG⁰ be a subgroup ofK^×satisfying

G^E < G⁰ < G . Then the multisets of d-parameters forGandG⁰coincide.

(5)

Proof. By [1, Corollary 16] it suffices to show that the following two finite abelian groups are isomorphic for every positive integern:

G

G∩(K^×)^`ⁿ and G⁰ G⁰∩(K^×)^`ⁿ .

By considering the mapG→G^Ewhich is raising to the powerEone can easily deduce that

(4) G

G∩(K^×)^`ⁿ ' G^E G^E∩(K^×)^`ⁿ .

The claim boils down to the fact that ifx∈Ganda∈K^×are such thatx^E =a^`ⁿ, thenxis an

`ⁿ-th power inK^×, which follows from the identityx= (a^c¹x^c²)^`ⁿ, wherec1E+c2`ⁿ= 1.

By considering the inclusionsG^E ,→G⁰ ,→Gwe get inclusions G^E

G^E ∩(K^×)^`ⁿ ,→ G⁰

G⁰∩(K^×)^`ⁿ ,→ G G∩(K^×)^`ⁿ

and we may conclude because of the isomorphism (4).

Remark 8. LetGbe a torsion-free subgroup ofK^×of rank2. LetEbe an integer coprime to

`, and letG⁰ be a subgroup ofK^×satisfying

G^E < G⁰ < G .

Then for every nonnegative integernwe have an equality of Kummer extensions:

K(ζ_`ⁿ, ^`n√

G) =K(ζ_`ⁿ, ^`n√ G⁰).

The proof is immediate from Kummer theory (in short: an inclusion between groups implies an inclusion of the corresponding Kummer extensions, and the Kummer extensions related toG andG^E are the same becauseEis coprime to`).

Proposition 9. LetAandB be elements ofK^×which are strongly independent, and letζ, ξ be roots of unity inµK,`^∞. Consider non-zero integersX, Y, Zand the elements

b₁ = A^Xζ b₂ = A^YB^Zξ .

CallGthe grouphb₁, b2iand writex, y, zfor the`-adic valuation ofX, Y, Z respectively.

(i) Ifx6y, then the d-parameters forGarexandz. Moreover there is a subgroupG⁰of K^×satisfying

K(ζ`ⁿ, ^`n

√

G) =K(ζ`ⁿ, ^`n

√ G⁰)

for all nonnegative integersnand such that the(d, h)-parameters forG⁰are

(5) (x,ord_`ζ) and (z,ord_`ξ⁰),

where ξ⁰ := ζ^Fξ^E withE, F non-zero integers such thatXF +Y E = 0andE is coprime to`.

(ii) Ifx > y, andz6y, then the(d, h)-parameters forGare

(6) (x,ord_`ζ) and (z,ord_`ξ).

(6)

(iii) Ifx > y, andz > y, then the d-parameters forGarex−y+zandy. Moreover there is a subgroupG⁰ ofK^×satisfying

K(ζ_`ⁿ, ^`n

√

G) =K(ζ_`ⁿ, ^`n

√ G⁰)

for all nonnegative integersnand such that the(d, h)-parameters forG⁰are (7) (x−y+z,ord_`ζ⁰) and (y,ord_`ξ),

whereζ⁰ := ζ^Eξ^F withE, F coprime (non-zero) integers such thatXE +Y F = 0 andEis coprime to`.

Proof. Case (i): Ifx 6 y, then there exists an integerE coprime to`and an integerF such thatXF +Y E= 0and hence

b1 = A^Xζ b^E₂b^F₁ = B^ZEξ⁰.

The groupG⁰=hb₁, bÊ₂b^F₁iis contained inGand it containsGÊso we deduce from Lemma 7 thatGhas the same d-parameters ofG⁰. By Lemma 6 these arexandv`(ZE) =z, while theh- parameters areord`ζ andord`ξ⁰. SinceGÊ < G⁰ < Gthe Kummer extensionsK(ζ`ⁿ, ^`n√

G) andK(ζ_`ⁿ, ^`n√

G⁰)coincide for everynby Remark 8.

Case (ii): We claim thatb1, b2are a good basis forG, so that the assertion follows from Remark 4 and Lemma 5. Ifb₁, b₂is not a good basis ofG, then by Remark 3 we can make a base change and increase the d-parameters of the basis. Sincex > ywe cannot increase the d-parameter of b₂ with a base change of the form

b_2,new =b^E₁b^F₂ =A^{XE+Y F}B^ZF ·ζ^Eξ^F

where the integersE, F are not both zero. Indeed,F has to be coprime to`becauseb₂ ∈G.

But then by Lemma 5 the divisibility parameter ofb_2,new is (sincex > yandF is coprime to

`)

min(v`(XE+Y F), v`(ZF)) = min(v`(Y), v`(Z)) and hence it is the same as the d-parameter ofb2.

Now we prove that we cannot increase the d-parameter ofb1with a base change of the form b_1,new =b^E₁b^F₂ =A^{XE+Y F}B^ZF ·ζ^Eξ^F

where the integersE, F are not both zero. The integerEmust be coprime to`in order to have a base change. Let us consider the d-parameter forb_1,new. In view of Lemma 5, we should havev`(ZF)> x. Moreover, in order to have

v_`(XE+Y F)> x=v_`(X) =v_`(XE)

we should havev_`(F) =x−y. So we havev_`(ZF) =z+x−y6x, contradiction.

Case (iii): We can find an integerEcoprime to`and an integerF such thatEX+Y F = 0 so we can write

b^E₁b^F₂ = B^ZF ·ζ^Eξ^F b₂ = A^YB^Zξ .

In particular, we must havev`(F) = x−y >0. By Case (ii), the d-parameters for the group G⁰ = hb^E₁b^F₂, b₂iarev_`(ZF) = x−y+zandy while the h-parameters areord_`ζ^Eξ^F and

(7)

ord`ξ. But the divisibility parameters ofG⁰andGare the same by Lemma 7, so we conclude

by Remark 8.

4. COMPUTINGKUMMER DEGREES

LetKbe a number field and`a prime number. Fix elementsα₁andα₂ofK^×which generate a torsion-free subgroup of K^× of rank 2. Let n1,n2 be non-negative integers, and set n :=

max(n₁, n₂). Letmbe an integer such thatm>n. The Kummer extension

(8) K(ζ_`^m, ^`n1√

α1, ^`n2√

α2)/K(ζ_`^m) is the same as the Kummer extension

K ζ_`^m, ^`n

q

(α₁)^`ⁿ⁻ⁿ¹, ^`n q

(α₂)^`ⁿ⁻ⁿ²

/K(ζ_`^m).

SettingG(n1, n2) :=h(α₁)^`ⁿ⁻ⁿ¹,(α2)^`ⁿ⁻ⁿ²i, this Kummer extension can be written as K ζ_`^m, ^`np

G(n1, n2)

/K(ζ_`^m).

Convention. For the remaining of the section (with the exception of Section 4.4), we suppose that`6= 2or thatζ4 ∈K.

Remark 10. By[1, Theorem 18], callingd_i := d_i(n₁, n₂)andh_i := h_i(n₁, n₂)fori= 1,2 the (d,h)-parameters ofG(n1, n2), the degree of (8)is a power of`with exponent

(9) max(h1+ min(n, d1)−m, h₂+ min(n, d2)−m,0) + max(n−d1,0) + max(n−d₂,0) so we are left to evaluate the (d,h)-parameters ofG(n₁, n₂).

Notice that ifh₁ =h₂= 0then the above formula simplifies to max(n−d₁,0) + max(n−d₂,0) (and this generalises straight-forwardly to higher rank).

Remark 11. Ifα1 andα2 are strongly independent, then by Lemma 6 we simply haved1 = n−n₁ andd₂ =n−n₂, whileh₁,h₂ may be taken to be zero, so the`-adic valuation of the Kummer degree(8)is simply

n1+n2.

This remark can be easily generalized to higher rank thus it is possible to prove(2)by applying [1, Theorem 18]under the assumption that`6= 2or thatζ4 ∈K.

By considering a good basis for the group hα₁, α2i we can find two strongly independent elementsA, BofK^×and roots of unityη, θ∈µ_K,`^∞and integersE₁, E₂, F₁, F₂such that we have

(10)

α₁ = A^E¹B^F¹η α2 = A^E²B^F²θ .

Notice that A, B, η, θ, E1, E2, F1, F2 are independent of n1, n2 and explicitly computable.

Writee₁, e₂, f₁, f₂for the`-adic valuation ofE₁, E₂, F₁, F₂ respectively (if such numbers are non-zero).

(8)

4.1. Special cases in which there are some zero exponents in(10). We now treat the case where some of the exponents in (10) are zero. Notice that there can be at most two zeros and that if there are two zeros, then we either haveE2 =F1 = 0orE1=F2 = 0. We first suppose that there are two zeros and apply Lemma 6:

•IfE2=F1 = 0, then the (d,h)-parameters forG(n1, n2)are

(n−n1+e1,ord_`η^`ⁿ⁻ⁿ¹) and (n−n2+f2,ord_`θ^`ⁿ⁻ⁿ²).

By plugging these values in (9) we get the `-adic valuation of the Kummer degree (8). If E1 =F2 = 0, we clearly have the analogous result.

Now we suppose that there is exactly one zero exponent and apply Proposition 9 (the integers E, F are as in Proposition 9 according to the case we are considering).

•IfF1 = 0, then to compute the Kummer degree(8)with(9)we can take as (d,h)-parameters:

(i) Ifn2−n1 6e2−e1:

(n−n₁+e₁,ord_`η^`ⁿ⁻ⁿ¹) and (n−n₂+f₂,ord_`η^{F `}ⁿ⁻ⁿ¹θ^E`ⁿ⁻ⁿ²). (ii) Ifn₂−n₁ > e₂−e₁andf₂ 6e₂:

(n−n1+e1,ord`η^`ⁿ⁻ⁿ¹) and (n−n2+f2,ord`θ^`ⁿ⁻ⁿ²). (iii) Ifn2−n1 > e2−e1andf2 > e2:

(n−n₁+e₁−e₂+f₂,ord_`η^E`ⁿ⁻ⁿ¹θ^{F `}ⁿ⁻ⁿ²) and (n−n₂+e₂,ord_`θ^`ⁿ⁻ⁿ²). IfE₁ = 0, or ifE₂ = 0, or ifF₂ = 0we have the analogous case distinction and result with respect to the caseF1= 0(it suffices to swapα1andα2, or swapAandB, or both).

4.2. The generic case in which there are no zero exponents in(10). SettingN₁ := `ⁿ⁻ⁿ¹ andN2 :=`ⁿ⁻ⁿ², we can write

α^N₁¹ = A^E¹^N¹B^F¹^N¹η^N¹ α^N₂² = A^E²^N²B^F²^N²θ^N².

Suppose thatv_`(F₁N₁) >v_`(F₂N₂). Then there are integersW₁, R₁ coprime to`(which are independent ofn₁, n₂and explicitly computable) such that we have

F₁N₁W₁+F₂N₂R₁`^(f¹⁺ⁿ⁻ⁿ¹^)−(f²⁺ⁿ⁻ⁿ²⁾= 0 so we get

(

α₁^N¹^W¹α₂^N²^R¹^`^(f¹⁻ⁿ¹⁾^−(f²⁻ⁿ²⁾ = A^E¹^N¹^W¹^+E²^N²^R¹^`^(f¹⁻ⁿ¹⁾^−(f²⁻ⁿ²⁾ν α^N₂² = A^E²^N²B^F²^N²θ^N²,

where

ν:= (η^W¹θ^R¹^`^f¹^−f²)^`ⁿ⁻ⁿ¹.

Notice thatνdepends onn₁andn₂only throughn₂−n₁ (becausen−n₁equalsn₂−n₁or 0), and (sinceνis the power of a computable root of unity) we can determineν with a finite

(9)

case distinction andν is explicitly computable in each of the finitely many cases. The above system can be rewritten as

(11)

(

α₁^N¹^W¹α₂^N²^R¹^`^(f¹⁻ⁿ¹⁾^−(f²⁻ⁿ²⁾ = A^S¹^N¹ν

α^N₂² = A^E²^N²B^F²^N²θ^N², where

S₁:=E₁W₁+E₂R₁`^f¹^−f².

Notice that the non-zero rational numberS1 is independent ofn1 andn2 and explicitly computable, and writes₁ :=v_`(S₁). SinceW₁ is coprime to`, by Remark 8 the groupG(n₁, n₂) has the same Kummer extensions as the group generated by the two elements in (11). Then we apply Proposition 9 to the latter group and get the following result:

•IfE₁, E₂, F₁, F₂ 6= 0andv_`(F₁N₁)>v_`(F₂N₂), then to compute the Kummer degree of (8) with(9)we can take as (d,h)-parameters (whereEandF are as in Proposition 9 according to the case distinction):

(i) Ifn₂−n₁ 6e₂−s₁:

(n−n1+s1,ord`ν) and (n−n2+f2,ord`ν^Fθ^`ⁿ⁻ⁿ²^E). (ii) Ifn₂−n₁ > e₂−s₁andf₂ 6e₂:

(n−n₁+s₁,ord_`ν) and (n−n₂+f₂,ord_`θ^`ⁿ⁻ⁿ²). (iii) Ifn2−n1 > e2−s1andf2 > e2:

(n−n₁+s₁−e₂+f₂,ord_`ν^Eθ^`ⁿ⁻ⁿ²^F) and (n−n₂+e₂,ord_`θ^`ⁿ⁻ⁿ²). Notice that the roots of unity in the formulas depend on n₁ andn₂, but only through a finite case distinction, so that all parameters that we will appear in formula (9) are explicitly computable.

IfE1, E2, F1, F2 6= 0andv`(F1N1) < v`(F2N2) one has the analogous result by swapping α₁andα₂.

We have considered all cases, and in particular have proven Theorem 2 under the assumption that`6= 2orζ₄∈K.

4.3. Formulas for the case ζ_` ∈/ K. Now suppose thatζ_` ∈/ K. With this assumption all h-parameters are zero and the formulas simplify.

Remark 12. Suppose that ζ` ∈/ K. Recall that e1, e2, f1, f2, s1 are integers independent fromn₁andn₂ and explicitly computable. Up to swappingα₁,α₂orA,Bor both, the`-adic valuation of the Kummer degree of (3)is given by one of the following expressions (according to the above case-distinction):

max(n₁−e₁,0) + max(n₂−f₂,0) max(n1−e1+e2−f2,0) + max(n2−e2,0)

max(n1−s1,0) + max(n2−f2,0) max(n₁−s₁+e₂−f₂,0) + max(n₂−e₂,0).

(10)

4.4. The case` = 2andζ4 ∈/ K. Now suppose that we are in the case` = 2andζ4 ∈/ K.

The Kummer extension

(12) K(ζ₂^m, ²ⁿ√¹

α₁, ²ⁿ√²

α₂)/K(ζ₂^m)

can be seen as a Kummer extension over the base fieldK(ζ4)as soon asn1 >2orn2 >2or m >2. If this holds, we can reduce to the previous case in whichζ4 belongs to the base field by simply extendingK toK(ζ₄)(notice that by the results in [5] the divisibility parameters overK(ζ4)can be deduced from the divisibility parameters overK). We are left to compute the degrees of the following extensions:

K(²ⁿ√¹

α₁, ²ⁿ√² α₂)/K

wheren1andn2are in{0,1}. Ifn1 = 0orn2= 0then this is trivial because we have at most to see if an element ofK^×is a square or not (the above degree can only be1or2in this special case). Now we are left to compute the degree of

K(√ α1,√

α2)/K .

By Kummer theory we only need to check whetherα₁ orα₂ orα₁α₂ are squares inK^×. In this way we can determine whether this degree is1,2, or4.

We have now dealt with the missing case`= 2andζ₄ ∈/ K, and in particular we have proven Theorem 2.

REFERENCES

[1] DEBRY, C. - PERUCCA, A.,Reductions of algebraic integers, J. Number Theory167(2016), 259–283.

[2] PALENSTIJN, W. J.,Radicals in arithmetic, PhD thesis, University of Leiden (2014), available onhttps:

//openaccess.leidenuniv.nl/handle/1887/25833.

[3] PERUCCA, A.,The order of the reductions of an algebraic integer, J. Number Theory,148(2015), 121–136.

[4] PERUCCA, A. - SGOBBA, P.,Kummer theory for number fields and the reductions of algebraic numbers, Int.

J. Number Theory,15(2019), 1617–1633.

[5] PERUCCA, A. - SGOBBA, P. - TRONTO, S.,Addendum to: Reductions of algebraic integers [J. Number Theory 167 (2016) 259–283], J. Number Theory209(2020), 391–395.

[6] PERUCCA, A. - SGOBBA, P. - TRONTO, S.,Explicit Kummer theory for the rationals, submitted for publication, preprint available on ORBiluhttps://orbilu.uni.lu/handle/10993/39282.

[7] PERUCCA, A. - SGOBBA, P. - TRONTO, S., Kummer theory for number fields via entanglement groups, submitted for publication, preprint available on ORBiluhttps://orbilu.uni.lu/handle/10993/

40831.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OFLUXEMBOURG, 6AV. DE LAFONTE, 4364 ESCH-SUR- ALZETTE, LUXEMBOURG

Email address:antonella.perucca@uni.lu