Power Integral Bases in Orders of Composite Fields

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Power Integral Bases in Orders of Composite Fields

István Gaál, Péter Olajos and Michael Pohst

CONTENTS 1. Introduction 2. Composite Fields 3. Proof of Theorem I 4. Applications Acknowledgements References

2000 AMS Subject Classification:Primary 11D57; Secondary 11R04 Keywords: Compositefields, power integral bases

We consider the existence of power integral bases in composites of polynomial orders of number fields. We prove that if the degree of the composite field equals the product of the degrees of its subfields and the minimal polynomials of the generating elements of the polynomial orders have a multiple linear factor in their factorization moduloq, then the composite order admits no power integral bases. As an application we provide several examples including a parametric family of “simplest sextic fields.”

1. INTRODUCTION

For any primitive elementα∈ZK theindex of αis defined as the module index

I(α) := (Z⁺K :Z⁺[α]).

Obviously, the discriminant and index ofαsatisfy D_K/_Q(α) =I(α)²DK,

whereDKis the discriminant of thefieldK. The element αgenerates apower integral basis{1,α, . . . ,αⁿ⁻¹} inK if and only ifI(α) = 1.

The problem of existence and construction of power integral bases in algebraic numberfields has been inten- sively studied in recent years; for a survey we refer to [Ga´al 99].

2. COMPOSITE FIELDS

Letf, g∈Z[x] be distinct monic irreducible polynomials (overQ) of degreesmandn, respectively. Letϕbe a root off and letψbe a root ofg. SetL=Q(ϕ),M =Q(ψ) and assume that the compositefieldK=LM has degree mn. We also assume that there is a prime numberq, such that bothf andg have a multiple linear factor (at least square) modq, that is, there existaf and ag in Zsuch that

f(a_f)≡f⁰(a_f)≡0 (mod q),

g(a_g)≡g⁰(a_g)≡0 (mod q). (2—1)

°c A K Peters, Ltd.

1058-6458/2001$0.50 per page Experimental Mathematics11:1, page 87

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88 Experimental Mathematics, Vol. 11 (2002), No. 1 Remark 2.1. Our assumption implies thatqdivides both the discriminant d(f) of the polynomial f and the discriminantd(g) ofg.

Remark 2.2. In [Ga´al 98] we considered fields that are composites of subfields with coprime discriminants. Ac- cording to the remark above in our case the fields we consider are composites of subfields whose discriminants are not coprime. This is the case in many interesting examples some of which we list at the end of the paper.

Consider the orderOf=Z[ϕ] of thefieldL, the order O^g=Z[ψ] of thefieldM and the composite orderO^{f g}= O^fO^g =Z[ϕ,ψ] in the composite field K =M L. Note that{1,ϕ, . . . ,ϕ^m⁻¹},{1,ψ, . . . ,ψⁿ⁻¹}and

{1,ϕ, . . . ,ϕ^m⁻¹,ψ,ϕψ, . . . ,ϕ^m⁻¹ψ,

. . . ,ψⁿ⁻¹,ϕψⁿ⁻¹, . . . ,ϕ^m⁻¹ψⁿ⁻¹}, areZ-bases ofO^f,O^g andO^{f g}, respectively.

Our main result is the following:

Proposition 2.3. Under the assumptions above the index of any primitive element of the orderO^{f g}is divisible byq.

As a consequence we have:

Proposition 2.4. Under the assumptions above the order O^{f g} has no power integral bases.

At the end of the paper we give several applications of the propositions.

Note that a similar phenomenon occurs for composite fields in other cases as well, cf. [Gaál 95], [Gaál 98], [Gaál 00].

3. PROOF OF PROPOSITION 1

Denote the conjugates of ϕ ∈ L by ϕ⁽ⁱ⁾ (1 ≤ i ≤ m) and the conjugates of ψ ∈ M by ψ^(j) (1 ≤ j ≤ n).

Denote by γ^(i,j) the conjugate of any element γ ∈ K under the automorphism mapping ϕ to ϕ⁽ⁱ⁾ and ψ to ψ^(j) (1≤i≤m,1≤j≤n).

The discriminants of the polynomialsf andg are d(f) = Y

1≤i<j≤m

(ϕ⁽ⁱ⁾−ϕ^(j))² d(g) = Y

1≤i<j≤n

(ψ⁽ⁱ⁾−ψ^(j))². (3—1) These are also the discriminants of the bases {1,ϕ, . . . ,ϕ^m⁻¹} of the order Of and {1,ψ, . . . ,ψⁿ⁻¹}

of the order O^g, respectively. The discriminant of the orderOf gis

D(Of g) =d(f)ⁿ·d(g)^m. (3—2) We can represent any element α∈Of g in the form

α=

mX−1

i=0 n−1

X

j=0

xijϕⁱψ^j (3—3) with x_ij ∈ Z. The index of α (generating K over Q) corresponding to the orderO^{f g} is defined to be the Z- module index ofZ[α] inOf g. It is

I_O_{f g}(α) = 1 p|D(Of g)|

Y

(i1,j1)<(i2,j2)

¯¯

¯α⁽ⁱ¹^,j¹⁾−α⁽ⁱ²^,j²⁾

¯¯

¯ where the pairs of indices are ordered lexicographically.

Now we rearrange the factors in the product above. Using (3—1) and (3—2) we have

I_O_{f g}(α) = Ym

i=1

Y

1≤j1<j2≤n

¯¯

¯¯α^(i,j¹⁾−α^(i,j²⁾ ψ^(j¹⁾−ψ^(j²⁾

¯¯

× Yn

j=1

Y

1≤i1<i2≤m

¯¯

¯¯α⁽ⁱ¹^,j)−α⁽ⁱ²^,j) ϕ⁽ⁱ¹⁾−ϕ⁽ⁱ²⁾

¯¯

× Y

(i1, j1)<(i2, j2) i16=i2

j16=j2

¯¯

¯α⁽ⁱ¹^,j¹⁾−α⁽ⁱ²^,j²⁾

¯¯

¯.

(3—4) Obviously, the factors that appear in (3—4) are algebraic integers.

For any 1≤i1< i2≤mand 1≤j1< j2≤nwe have

³

α⁽ⁱ¹^,j¹⁾−α⁽ⁱ²^,j¹⁾

´ +

³

α⁽ⁱ²^,j¹⁾−α⁽ⁱ²^,j²⁾

´ +

³

α⁽ⁱ²^,j²⁾−α⁽ⁱ¹^,j¹⁾

´

= 0 which implies the equation

³

ϕ⁽ⁱ¹⁾−ϕ⁽ⁱ²⁾´ ε+³

ψ^(j¹⁾−ψ^(j²⁾´

η+ρ= 0 (3—5) with

ε= α⁽ⁱ¹^,j¹⁾−α⁽ⁱ²^,j¹⁾ ϕ⁽ⁱ¹⁾−ϕ⁽ⁱ²⁾ , η= α⁽ⁱ²^,j¹⁾−α⁽ⁱ²^,j²⁾

ψ^(j¹⁾−ψ^(j²⁾ , ρ=α⁽ⁱ²^,j²⁾−α⁽ⁱ¹^,j¹⁾.

Since these elements are factors in (3—4) they are algebraic integers lying in the Z-order O = Oⁱ1,i2,j1,j2 = Z[ϕ⁽ⁱ¹⁾,ϕ⁽ⁱ²⁾,ψ^(j¹⁾,ψ^(j²⁾].

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Ga´al et al.: Power Integral Bases in Orders of Composite Fields 89 Let us fix those indices 1 ≤ i1 < i2 ≤ m and 1 ≤

j₁ < j₂ ≤ n for which ϕ⁽ⁱ¹⁾ ≡ ϕ⁽ⁱ²⁾ (mod q) and also ψ^(j¹⁾ ≡ ψ^(j²⁾ (mod q). Consider equation (3—5) modulo q.

By our assumptions ϕ⁽ⁱ¹⁾−ϕ⁽ⁱ²⁾ ≡ 0 (mod q) and ψ^(j¹⁾−ψ^(j²⁾≡0 (mod q), hence by equation (3—5) we get ρ=α⁽ⁱ²^,j²⁾−α⁽ⁱ¹^,j¹⁾ ≡0 (mod q). This is one of the algebraic integer factors ofI(α), hence q|I(α).

4. APPLICATIONS 4.1 A Cyclic Sextic Field

Consider the sexticfieldKgenerated by a root ofh(x) = x⁶−x⁵−6x⁴+ 6x³+ 8x²−8x+ 1. This is a totally real cyclic sexticfield with discriminantD_K= 453789 = 3³7⁵. Its cubic subfield isL=Q(ϕ) (with discriminant 49) whereϕis a root of f(x) =x³+ 4x²+ 3x−1. In the fieldLthe elements{1,ϕ,ϕ²}form an integral basis. We have f(x)≡(x+ 6)³ (mod 7). The quadratic subfield isM =Q(√

21). The polynomialg(x) =x²−x−5 has ψ = (1 +√

21)/2 as a root, and obviously {1,ψ} is an integral basis in M. We have g(x) ≡(x−1/2)² (mod 7). Proposition 1 implies that the indices of the primitive elements of the orderOf g=Z[1,ϕ,ϕ²,ψ,ϕψ,ϕ²ψ] are all divisible by 7, hence it has no power integral basis.

4.2 A Non-Cyclic Sextic Field

Consider the sextic field K generated by a root of h(x) =x⁶−12190x⁴+ 256565x²−12167. This is a totally real sexticfield with Galois groupD6, discriminant DK = 2⁶17²23³647². Its cubic subfield isL=Q(ϕ) (with discriminant 252977 = 17·23·647 and Galois groupS3) whereϕis a root off(x) =x³−22x²−23x−1. In the fieldLthe elements{1,ϕ,ϕ²}form an integral basis. We havef(x)≡(x+15)(x+ 16)² ( mod 23). The quadratic subfield isM =Q(√

23). The polynomialg(x) =x²−23 hasψ=√

23 as a root, and obviously{1,ψ}is an integral basis in M. We haveg(x)≡x² (mod 23). Proposition 1 implies that the indices of the primitive elements of the order Of g = Z[1,ϕ,ϕ²,ψ,ϕψ,ϕ²ψ] are all divisible by 23, hence it has no power integral basis.

4.3 The Parametric Family of Simplest Sextic Fields Assume 3-t, t6=−8,±5. Let us consider the family of sexticfieldsKt generated by a root β of the polynomial

ht(x) =x⁶−2tx⁵−(5t+ 15)x⁴−20x³ + 5tx²+ (2t+ 6)x+ 1.

This family offields is called the “simplest sexticfields”.

It has some attractive properties which are listed in [Lettl et al. 98]. Thesefields are totally real cyclicfields. Letp be a prime dividingq=t²+3t+9. We haved(h_t) = 6⁶q⁵. Note thatht(x)≡ (x−t/3)⁶ (mod p) (the “simplest quintic fields” have a similar property, cf. [Ga´al and Pohst 97]).

The cubic subfield Lt ofKt is generated by a rootϕ off_t =x³−tx²−(t+ 3)x−1 withd(f_t) =q². These are the “simplest cubicfields”, totally real, cyclic. It is well known that {1,ϕ,ϕ²} is an integral basis ofZ[ϕ]. Note thatft(x)≡(x−t/3)³ (mod p).

The quadratic subfield ofKt isMt =Q(√q) . If q ≡ 2,3 (mod 4) then set gt(x) = x² −q with d(gt) = 4q and with a root ψ = √q. In this case g_t(x)≡x² (mod p).

Ifq ≡1 (mod 4) then set gt(x) = x²−x−(q−1)/4 withd(g_t) =qand with a rootψ= (1 +√q)/2. In this casegt(x)≡(x−1/2)² (mod p).

In both cases{1,ψ}is an integral basis ofMt.

Consider now the orderOf g=Z[1,ϕ,ϕ²,ψ,ϕψ,ϕ²ψ].

By Proposition 1 the indices of the primitive elements of Of g are all divisible by p, hence Of g has no power integral bases.

4.4 A Field of Higher Degree

This is an example to illustrate that our results are easily applicable also to suitablefields of higher degrees.

Let ϕ be a root of f(x) = x⁵−2x⁴+ 7x²+ 6x+ 5.

The quinticfield L=Q(ϕ) has no non-trivial subfields.

Let ψ be a root of g(x) = x⁸+ 13x⁷+ 55x⁶+ 75x⁵ + 2x³−x²−143x−525. The octicfieldM =Q(ψ) has no non-trivial subfields, either. We have

f(x)≡(x+ 16)²(x³+ 16x+ 5) (mod 17) g(x)≡(x+ 5)²(x³+ 12x²+ 2x+ 14)

×(x³+ 8x²+ 4x+ 7) (mod 17)

hence our Proposition 1 applies. Consider the order O^{f g} = Z[ϕ,ψ] of the field K = Q(ϕ,ψ) of degree 40.

Anyα∈Of g can be represented in the form

α= X4

i=0

X7

j=0

xijϕⁱψ^j

withxij∈Z. By Proposition 1 the indices of all primitive elements ofO^{f g}are divisible by 17, henceO^{f g}admits no power integral bases.

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90 Experimental Mathematics, Vol. 11 (2002), No. 1

ACKNOWLEDGEMENTS

The research of István Gaál was supported in part by Grants T 29330 and T 037367 from the Hungarian National Foun- dation for Scientific Research and by FKFP 0343/2000. The research of Péter Olajos was supported by FKFP 0343/2000.

REFERENCES

[Ga´al 95] I.Ga´al. “Computing elements of given index in totally complex cyclic sexticfields.”J. Symbolic Computa- tion20(1995), 61—69.

[Ga´al 98] I.Ga´al. “Power integral bases in composits of numberfields.”Canad. Math. Bulletin41(1998), 158—165.

[Ga´al 99] I.Ga´al. “Power integral bases in algebraic number fields.” Annales Univ. Sci. Budapest., Sect. Comp. 18 (1999), 61—87.

[Ga´al 00] I.Ga´al. “Solving index form equations in fields of degree nine with cubic subfields.”J. Symbolic Comput.

30(2000), 181—193.

[Ga´al and Pohst 97] I.Ga´al and M.Pohst. “Power integral bases in a parametric family of totally real quintics.”

Math. Comp.66(1997), 1689—1696.

[Lettl et al. 98] G.Lettl, A.Peth˝o, P.Voutier. “On the arith- metic of simplest sextic fields and related Thue equations.” In Number Theory, eds. K.Gy˝ory, A.Peth˝o, V.T.S´os, pp. 331—348, Walter de Gruyter, Berlin-New York, 1998.

István Gaál, University of Debrecen, Mathematical Institute H-4010 Debrecen Pf.12., Hungary ([email protected]) Péter Olajos, University of Debrecen, Mathematical Institute H-4010 Debrecen Pf.12., Hungary ([email protected]) Michael Pohst, Technische Universität Berlin, Fakultät II, Institut für Mathematik, Strasse des 17. Juni 136,

Berlin 10623, Germany ([email protected])

Received February 28, 2001; accepted in revised form August 14, 2001.