Constructing totally p-adic numbers of small height
aa r X i v : . [ m a t h . N T ] J a n CONSTRUCTING TOTALLY p -ADIC NUMBERS OF SMALL HEIGHT S. CHECCOLI AND A. FEHM
Abstract.
Bombieri and Zannier gave an effective construction of algebraic numbersof small height inside the maximal Galois extension of the rationals which is totallysplit at a given finite set of prime numbers. They proved, in particular, an explicitupper bound for the lim inf of the height of elements in such fields. We generalize theirresult in an effective way to maximal Galois extensions of number fields with given localbehaviour at finitely many places. Introduction
Let h denote the absolute logarithmic Weil height on the field Q of algebraic numbers.We are interested in explicit height bounds for elements of Q with special local behaviourat a finite set of primes. The first result in this context is due to Schinzel [Sch73] whoproved a height lower bound for elements in the field of totally real algebraic numbers Q tr , the maximal Galois extension of Q in which the infinite prime splits totally. Moreprecisely, he showed that every α ∈ Q tr has either h ( α ) = 0 or h ( α ) ≥
12 log √ ! . Explicit upper and lower bounds for the limit infimum of the height of algebraic integersin Q tr are given in [Smy80, Smy81, Fla96].In [BZ01] Bombieri and Zannier investigate the analogous problem for the p -adic num-bers. More precisely, in [BZ01, Theorem 2] they prove the following: Theorem 1.1 (Bombieri–Zannier) . Let p , . . . , p n be distinct prime numbers, for each i let E i be a finite Galois extension of Q p i , and L the maximal Galois extension of Q contained in all E i . Denote by e i and f i the ramification index and inertia degree of E i / Q p i . Then lim inf α ∈ L h ( α ) ≥ · n X i =1 log( p i ) e i ( p f i i + 1) . In the special case E i = Q p i , Bombieri and Zannier in [BZ01, Example 2] show thatthe lower bound in Theorem 1.1 is almost optimal. More precisely: Theorem 1.2 (Bombieri–Zannier) . Let p , . . . , p n be prime numbers and let L be themaximal Galois extension of Q contained in all Q p i . Then lim inf α ∈ L h ( α ) ≤ n X i =1 log( p i ) p i − . Other proofs, refinements and generalizations were given in [Fil14, Pot15, FP15, FP19,PS19] See also [Smy07] for a general survey on the height of algebraic numbers. In Remark1.4 we will discuss in detail the contribution [Fil14] and how it compares to our work. he goal of this note is to generalize in an effective way the upper bound Theorem 1.2to general E i , and to further replace the base field Q by an arbitrary number field. Ourmain result is the following: Theorem 1.3.
Let K be a number field and let p , . . . , p n be distinct primes ideals of thering of integers O K of K . For each i , let E i be a finite Galois extension of the completion F i of K at p i . Denote by e i and f i the ramification index and the relative inertia degreeof E i /F i and write q i = |O K / p i | = p f ( p i | p i ) i . Then for the maximal Galois extension L of K contained in all E i , lim inf α ∈ L h ( α ) ≤ n X i =1 f ( p i | p i )[ K : Q ] · log( p i ) e i ( q f i i − . More precisely, let C = max n [ K : Q ] , | ∆ K | , max i ( e i f i ) , max i q f i i o where ∆ K is the absolute discriminant of K . Then for every < ǫ < there existinfinitely many α ∈ O L of height (1.1) h ( α ) ≤ n X i =1 f ( p i | p i )[ K : Q ] · log( p i ) e i ( q f i i −
1) + 13 nC n +2 log ([ K ( α ) : K ])[ K ( α ) : K ] + ( , n = 1 nǫ, n > . Namely, for every ρ ≥ C n there exists such α of degree (1.2) ρ ≤ [ K ( α ) : K ] ≤ ( Cρ, n = 1 ρ (4 log C ) n +1log n (1+ ǫ ) , n > . Note that in the special case K = Q and E i = Q p i we reobtain Theorem 1.2, exceptthat Theorem 1.3 appears stronger in that the result is effective and the lim inf canbe taken over algebraic integers. However, an inspection of the proof of Bombieri andZannier shows that it is effective as well and does in fact produce algebraic integers. Remark . Theorem 1.3 provides an effective version of a result of Fili [Fil14, Theorem1.2]. The bound in that result seems to differ from ours by the factor e ( p i | p i ), and [Fil14,Theorem 1.1] (and similarly [FP15, Theorem 9]) also states a variant of Theorem 1.1which contradicts our Theorem 1.3, but according to Paul Fili (personal communication)this is merely an error in normalization in [Fil14] and [FP15] that became apparentwhen comparing to our result, and the e v in the denominator of Theorems 1.1, 1.2, andConjecture 1 of [Fil14] (and similarly in the statements of [FP15]) should have been theabsolute instead of the relative ramification index. When this correction is made, thelower bound of [Fil14, Theorem 1.1] agrees with the one in [FP19, Theorem 13], and theupper bound of [Fil14, Theorem 1.2] agrees with the one in Theorem 1.3.In any case, Fili’s proof of [Fil14, Theorem 1.2] uses capacity theory on analyticBerkovich spaces and does not provide explicit bounds on the degree and the heightof a sequence of integral elements in the lim inf. Instead, our effective proof is moreelementary and is inspired by Bombieri and Zannier’s effective proof of Theorem 1.2. Tothe best of our knowledge, Theorem 1.3 is the only result currently available that gives abound on the height in terms of the degree of such a sequence of α , except for the casewhere K = Q and E i = Q p i for all i , where such a bound can be deduced from [BZ01].We also remark that our use of [Fil14, Theorem 1.2] in [CF21] is limited to the caseswhere [Fil14, Theorem 1.2] agrees with Theorem 1.3. he paper is organised as follows. In Section 2 we collect all the preliminary resultsneeded to prove Theorem 1.3, namely: a consequence of Dirichlet’s theorem on simulta-neous approximation (Proposition 2.1), a bound for the size of representatives in quotientrings of rings of integers (Proposition 2.2), a variant of Hensel’s lemma (Proposition 2.3),a bound for the height of a root of a polynomial defined over a number field in termsof its coefficients (Proposition 2.4), and a construction of special Galois invariant sets ofrepresentatives of residue rings of local fields (Proposition 2.5).The proof of Theorem 1.3 is carried out in Section 3. We briefly sketch it here forclarity. Following Bombieri and Zannier’s strategy, given ρ ≥ C n we construct a monicirreducible polynomial g ∈ O K [ X ] such that(i) its degree is upper and lower bounded in terms of ρ as the degree of α in Theorem 1.3,(ii) the complex absolute value of all conjugates of its coefficients is sufficiently small,and(iii) all its roots are contained in all E i .In Bombieri and Zannier’s proof of Theorem 1.2, (i) and (ii) were achieved by using theChinese Remainder Theorem to deform the polynomial Q ρi =1 ( X − i ) into an irreduciblepolynomial of the same degree with coefficients small enough to give the desired boundfor the height of the roots. Then a variant of Hensel’s lemma was applied to show thatthe roots of the constructed polynomial are still in Q p i for each i .In our generalisation, the degree of the polynomial is carefully chosen to obtain (i) inSection 3.1 via Proposition 2.1 (necessary only if n >
1, which leads to the better boundsin the case n = 1). The polynomial g satisfying (ii) is then constructed in Section 3.2:We start with polynomials Q α ∈ ˜ A i ( X − α ), where now ˜ A i ⊆ O E i is a set constructed usingProposition 2.5. These polynomials are then merged into an irreducible polynomial g by applying the Chinese Remainder Theorem and Proposition 2.2 to bound the size ofits coefficients. Property (iii) is verified in Section 3.3, using Proposition 2.3. Finally,Proposition 2.4 is applied to show that g has a root α of height bounded from above asdesired. 2. Notation and preliminaries
We fix some notation. If K is a number field or a non-archimedean local field we let O K denote the ring of integers of K . For an ideal a of O K we denote by N ( a ) = |O K / a | itsnorm. For a nonzero prime ideal p of O K , we denote by v p the discrete valuation on K with valuation ring ( O K ) p normalized such that v p ( K × ) = Z . If L/K is an extension ofnumber fields and P is a prime ideal of O L lying above a prime ideal p of O K we denoteby e ( P | p ) and f ( P | p ) the ramification index and the inertia degree. For an extension E/F of non-archimedean local fields we denote the ramification index and the inertiadegree also by e ( E/F ) and f ( E/F ).2.1.
Auxiliary results.
In this section we collect the preliminary results we need toprove Theorem 1.3. These results are not related to each other and we list them in thissection following their order of appearance in the proof of Theorem 1.3.
Proposition 2.1.
Let x , . . . , x n be integers greater than . For every ρ ≥ and < ǫ < there exist positive integers r, k , . . . , k n such that r ≥ ρ and, for all i , r ≤ x k i i ≤ (1 + ǫ ) r and k i ≤ n +1 log n (max j x j ) log( ρ )log( x i ) log n (1 + ǫ ) . roof. Say x = max i x i . Let α i = 2 log( ρ ) / log( x i ) and Q = ⌈ x ) / log(1 + ǫ ) ⌉ . Bythe simultaneous Dirichlet approximation theorem [Sch80, Chapter II, Section 1, Theorem1A] there exist positive integers q, k , . . . , k n with 1 ≤ q < Q n such that | qα i − k i | ≤ Q − for all i , and thus | ρ ) q − log( x k i i ) | ≤ log(1 + ǫ ) /
2. Letting r = min i x k i i , one haslog( r ) ≥ ρ ) q − ≥ log( ρ ) for q ≥ ρ ≥
3. In addition, for all i , 0 ≤ log( x k i i ) − log( r ) ≤ log(1 + ǫ ), hence r ≤ x k i i ≤ (1 + ǫ ) r . Finally k i ≤ qα i + 1 ≤ qα i ≤ ρ ) Q n log( x i ) − and replacing Q we get the desired bound. (cid:3) The next proposition deals with bounds for the absolute value of small representativesfor quotient rings.
Proposition 2.2.
Let K be a number field of degree m = [ K : Q ] . Given a nonzero ideal a of O K , there exists a set of representatives A of O K / a such that, for every a ∈ A andevery σ ∈ Hom( K, C ) , one has | σ ( a ) | ≤ δ K N ( a ) /m where δ K = m m ( m − p | ∆ K | .Proof. This is an immediate consequence of well-known results on lattice reduction. Forinstance, [BFH17, Proposition 15] gives for every α ∈ O K , an element a ∈ O K with α − a ∈ a such that sX σ | σ ( a ) | ≤ m ℓ m ( m − p | ∆ K | N ( a ) /m where ℓ depends on certain parameters η ∈ (1 / , , δ ∈ ( η ,
1) and θ > §
4, p.595]) to a Z -basis of a . In particular,choosing η = 2 / , δ = 7 / θ = ( √ − /
3, we have ℓ = 2, which gives the claimedupper bound for | σ ( a ) | . (cid:3) The following proposition is a variant of Hensel’s lemma.
Proposition 2.3.
Let E be a finite extension of Q p , P the maximal ideal of O E and v = v P . Let f ∈ E [ X ] and x ∈ E . Assume there exist a, b ∈ Z such that(i) v ( f ( x )) > a + b ,(ii) v ( f ′ ( x )) ≤ a ,(iii) v ( f ( ν ) ( x ) /ν !) ≥ a − ( ν − b for every ν ≥ .Then there exists x ∈ E with f ( x ) = 0 and v ( x − x ) > b .Proof. This can be proved precisely as the special case E = Q p in [BZ01, Lemma 1].Alternatively, one can reduce this to one of the standard forms of Hensel’s lemma asfollows. Let β ∈ E with v ( β ) = b . Then g ( X ) := ( βf ′ ( x )) − f ( βX + x ) is in O E [ X ] by( i )-( iii ) and has a simple zero X = 0 modulo P by ( i ) and ( ii ), hence by Hensel’s lemma g has a zero x ′ ∈ P , and x = βx ′ + x is then the desired zero of f . (cid:3) The final proposition in this subsection gives a bound for the height of the roots of apolynomial with small algebraic coefficients.
Proposition 2.4.
Let K be a number field and let f ( X ) = X m + a m − X m − + . . . + a ∈O K [ X ] . If B ≥ with | σ ( a i ) | < B for every i and every σ ∈ Hom( K, C ) , then f has aroot α with h ( α ) ≤ log( B √ m + 1) m . roof. Let M K = M K ∪ M ∞ K be the set of (finite and infinite) places of K and let d = [ K : Q ]. For a place v ∈ M K , denote by d v = [ K v : Q v ] the local degree. Letˆ h ( f ) = log Y v ∈ M K M v ( f ) d v /d ! where if v is non-archimedean M v ( f ) = max i ( | a i | v ), while if v is archimedean and corre-sponds to the embedding σ ∈ Hom( K, C ), M v ( f ) is the Mahler measure M ( σ ( f )) of thepolynomial σ ( f ).By [Zan09, Appendix A, section A.2, pag. 210] we have that, if α , . . . , α m ∈ Q arethe roots of f (with multiplicities), then ˆ h ( f ) = P mi =1 h ( α i ) where h denotes the usuallogarithmic Weil height. Thus, if α is a root of f of minimal height, then(2.1) h ( α ) ≤ ˆ h ( f ) m . Let σ , . . . , σ r and τ , τ , . . . , τ s , τ s be, respectively, the real and pairwise conjugate com-plex embeddings of K in C , so that d = r + 2 s . As f has coefficients in O K , M v ( f ) ≤ v is non-archimedean and we have thatˆ h ( f ) ≤ log Y v ∈ M ∞ K M v ( f ) d v /d = log r Y i =1 M ( σ i ( f )) /d · s Y j =1 M ( τ j ( f )) /d ! == log Y σ ∈ Hom( K, C ) M ( σ ( f )) /d . By [Zan09, Section 3.2.2, formula (3.7)] and by our hypothesis on B , we have that M ( σ ( f )) ≤ B √ m + 1 for all σ ∈ Hom( K, C ), thus ˆ h ( f ) ≤ log( B √ m + 1) and, pluggingthis bound into (2.1), we conclude. (cid:3) Representatives of residue rings of local fields.
This subsection contains thetechnical key result needed to construct the local polynomials in the proof of Theorem1.3. Let
E/F be a Galois extension of non-archimedean local fields with Galois group G .Let p be the maximal ideal of O F , P be the maximal ideal of O E and for k ∈ N denoteby π k : O E → O E / P k the residue map.It is known that one can always find a G -invariant set of representatives of the residuefield O E / P , e.g. the Teichm¨uller representatives. As long as the ramification of E/F istame, one can also find G -invariant sets of representatives of each residue ring O E / P k ,but if the ramification is wild, this is not necessarily so. We will therefore work with thefollowing substitute for such a G -invariant set of representatives: Proposition 2.5.
Let
E/F be a Galois extension of non-archimedean local fields anddefine G, p , P , π k as above. Let d be a multiple of | G | . There exists a constant c such thatfor every k there is A ⊆ O E such that(1) A is G -invariant,(2) all orbits of A have length | G | ,(3) π k | A : A → O E / P k is d -to-1 and onto, and(4) π k + c | A is injective.Moreover, if F is a p -adic field, one can choose c ≤ e ( P | p ) (cid:18) d + | G | + e ( p | p ) p − (cid:19) . roof. Note that G naturally acts on O E and on O E / P k , and that π k is G -equivariant.Fix some primitive element α ∈ O × E of E/F and a uniformizer θ ∈ O F of v p , and let e = e ( P | p ) ,c = max = σ ∈ G v P ( α − σα ) , (with c = 0 if G = 1) ,c = ⌈| G | + c /e ⌉ , and c = e ( d + c ) . Let k ∈ N be given. The desired set A is obtained by applying the following Claim inthe case X = O E / P k : Claim.
For every G -invariant subset X ⊆ O E / P k there exists a G -invariant subset A ⊆O E with all orbits of length | G | such that π k + c | A is injective and π k | A is d -to-1 onto X .We prove the Claim by induction on | X | : If X = ∅ , A = ∅ satisfies the claim. If X = ∅ take x ∈ X and let X ′ = X \ Gx , where Gx denotes the orbit of x under G . By theinduction hypothesis there exists A ′ ⊆ O E satisfying the claim for X ′ . Choose a ∈ π − k ( x )and let k = ⌈ ke ⌉ . Then n := min { n ≥ v P ( a − σa ) = e ( k + n ) + v P ( α − σα ) ∀ = σ ∈ G } < | G | , as v P ( a − σa ) − v P ( α − σα ) attains less than | G | many distinct values. Thus for 1 = σ ∈ G , v P (( a + θ n + k α ) − σ ( a + θ n + k α )) = min { v P ( a − σa ) , e ( n + k ) + v P ( α − σα ) }≤ e ( n + k ) + c < k + ec , so if we replace a by a + θ n + k α , we can assume without loss of generality that π k + ec isinjective on Ga and that | Ga | = | G | . If we now let A = A ′ ∪ { σ ( a ) + θ k + c + j : σ ∈ G, ≤ j < d/ | G x |} where G x is the stabilizer of x , then π k | A is d -to-1 onto X = X ′ ∪ Gx and A is G -invariantwith all orbits of length | G | . As k + ec ≤ e ( k + c + j ) < k + c, we have that π k + c | A is injective.Now, if F is a p -adic field and if we chose α ∈ O E to be also a generator of O E as a O F -algebra, by [Ser80, Chap.IV, Ex. 3(c)], one has the explicit bound c ≤ e ( P /p ) / ( p − c . (cid:3) Remark . Note that if (4) holds for some c, k, A , then also for c ′ , k, A for any c ′ ≥ c .3. Proof of Theorem 1.3
Using the notation of Theorem 1.3, for every 1 ≤ i ≤ n , let P i be the maximal ideal of O E i , v i the extension of v P i to an algebraic closure of E i , G i = Gal( E i /F i ) and d = Q ni =1 | G i | .Let C be the constant from Theorem 1.3, note that C ≥
2, and let c = 4 C n +1 . Fix aninteger ρ ≥ C n and note that ρ/d ≥ d ≤ C n . If n = 0 let ǫ = 0, otherwise fix0 < ǫ < .1. Choosing the right degree. If n > x i = q if i toobtain positive integers r > ρ/d and k , . . . , k n such that for every i ,(i) r ≤ q f i k i i ≤ (1 + ǫ ) r and(ii) k i ≤ n +1) (log C ) n log( ρ/d )log n (1+ ǫ ) ,where we used that, for every i , log 2 ≤ log( x i ) = log( q f i i ) ≤ log C . It follows thatlog( ρ/d ) ≤ log( r ) ≤ (4 log C ) n +1 log( ρ/d )log n (1 + ǫ ) . If n = 1 we instead set r = q f k , where k = ⌈ log( ρ/d ) / log( q f ) ⌉ , so that (i) holds with ǫ = 0, and log( ρ/d ) ≤ log( r ) ≤ log( ρ/d ) + log( q f ) . Using that (4 log C ) n +1 ≥ log n (1 + ǫ ), we conclude(3.1) ρ ≤ dr ≤ ( Cρ, n = 1 ρ (4 log C ) n +1log n (1+ ǫ ) , n > Construction of the polynomial g . We first want to prove the following:
Claim.
For every i , there exists a polynomial g i ∈ O K [ X ] of degree dr whose set of roots A i satisfies(a) A i ⊆ E i ,(b) v i ( α − β ) < k i + c for all α, β ∈ A i with α = β , and(c) v ( g i ′ ( α )) ≤ d (cid:18) q fikii − q fii − + c (cid:19) for every α ∈ A i . Proof of the claim. As e ( P i | p i )( d + | G i | + e ( p i | p i ) p i − + 1) ≤ C ( C n + C + C + 1) ≤ C n +1 = c ,by Proposition 2.5 there is a G i -invariant set A ′ i ⊆ O E i with all orbits of length | G i | suchthat A ′ i → O E i / P k i i is d -to-1 and A ′ i → O E i / P k i + ci is injective. As | A ′ i | = dq f i k i i , | G i | divides d , and r ≤ q f i k i i , there exists a G i -invariant subset ˜ A i ⊆ A ′ i with | ˜ A i | = dr . Let˜ g i = Y α ∈ ˜ A i ( X − α ) ∈ O F i [ X ] . We first prove that conditions (a)-(c) hold for ˜ g i and the set ˜ A i , instead of g i and A i . Notethat ˜ g i ∈ O F i [ X ] is monic of degree dr and that condition (a) holds for ˜ A i by construction.Moreover, as the map ˜ A i → O E i / P k i + ci is injective, we have that condition (b) is alsosatisfied for ˜ A i . As for condition (c), note that the valuation v P i on O E i induces a map¯ v : ( O E i / P k i i ) \ { } → { , . . . , k i − } such that v P i ( γ ) = ¯ v ( π k i ( γ )) for all γ ∈ O E i \ P k i i , here π k i denotes the residue map O E i → O E i / P k i i . Now v P i ( ˜ g i ′ ( α )) = X α = β ∈ ˜ A i v P i ( α − β ) ≤ X α = β ∈ A ′ i v P i ( α − β )= X α = β ∈ A ′ i π ki ( α )= π ki ( β ) v P i ( α − β ) + X α = β ∈ A ′ i π ki ( α ) = π ki ( β ) v P i ( α − β ) < ( d − · ( k i + c ) + d · X = a ∈O Ei / P kii ¯ v ( a )and X = a ∈O Ei / P kii ¯ v ( a ) = k i − X j =0 |{ a : ¯ v ( a ) = j }| · j = k i − X j =0 j X l =1 |{ a : ¯ v ( a ) = j }| = k i − X l =1 k i − X j = l |{ a : ¯ v ( a ) = j }| = k i − X l =1 |{ a : ¯ v ( a ) ≥ l }| = k i − X l =1 ( q f i ( k i − l ) i −
1) = k i − X l =0 q f i li − k i = 1 − q f i k i i − q f i i − k i and plugging this into the previous inequality gives condition (c) for ˜ g i .As O K is dense in O F i with respect to v i , we obtain a monic polynomial g i ∈ O K [ X ] ofdegree dr arbitrarily close to ˜ g i . Let A i be the set of roots of g i . By the continuity of roots[EP05, Theorem 2.4.7] we can achieve that the roots of g i are arbitrarily close to the rootsof ˜ g i , in particular that conditions (b) and (c) are satisfied by g i and A i . Moreover, byKrasner’s lemma [Lan94, Ch.II, §
2, Proposition 4], we can in addition achieve condition(a), completing the proof of the claim. (cid:3)
Now, let p be the smallest prime number not in the set { p , . . . , p n } and let p be aprime ideal of O K above p . Fix a monic polynomial g ∈ O K [ X ] of degree dr whosereduction modulo p is irreducible. Let(3.2) m i = de i q f i k i i − q f i i − k i + 2 c ! and a = p p m · · · p m n n . By the Chinese Remainder Theorem and Proposition 2.2 there exists a monic polynomial g ∈ O K [ X ] such that(1) deg g = dr ,(2) g ≡ g mod p [ X ],(3) g ≡ g i mod p m i i [ X ] for i = 1 , . . . , n , and(4) | σ ( a ) | ≤ δ K N ( a ) / [ K : Q ] for every coefficient a of g and every σ ∈ Hom( K, C ),where δ K = [ K : Q ] [ K : Q ]([ K : Q ] − p | ∆ K | .Note that (2) implies that g is irreducible. In particular, we get from (1) and (3.1)that every root α of g satisfies the degree bound (1.2) of Theorem 1.3. .3. The roots of g are in E i for every i . We claim that the conditions of Proposition2.3 hold for the field E i , the polynomial g and x = α ∈ A i (which lies in E i by condition(a)) by setting a = v i ( g i ′ ( α )) and b = k i + c −
1. Indeed, by (c) a = v i ( g i ′ ( α )) ≤ d · q f i k i i − q f i i − dc < e i m i − b, (3.3)and, writing g − g i = t i with t i ∈ p m i i [ X ] by (3) of Section 3.2, we have g ( α ) = t i ( α ) andtherefore v i ( g ( α )) ≥ e i m i > a + b , so condition (i) holds. Similarly for condition (ii), wehave g ′ ( α ) = t ′ i ( α ) + g i ′ ( α ) and since v i ( t ′ i ( α )) ≥ e i m i > v i ( g i ′ ( α )) , we conclude that v i ( g ′ ( α )) = v i ( g i ′ ( α )) = a . Now for ν ≥ g i ( ν ) ( α ) = ν ! g i ′ ( α ) X B ⊆ A i | B | = ν − α/ ∈ B Y β ∈ B ( α − β ) − . Thus v i ( g i ( ν ) ( α ) /ν !) ≥ a − ( ν −
1) max β = α v i ( α − β ) ≥ a − ( ν − b where the last inequality holds by (b). Moreover, using (3.3), we get v i ( t ( ν ) i ( α ) /ν !) ≥ e i m i − v i ( ν !) ≥ a + b − e ( P i | p i ) νp i − ≥ a − ( ν − b where the last inequality holds since b ≥ c ≥ e ( P i | p i ) p i − . Thus v i ( g ( ν ) ( α ) /ν !) ≥ min n v i ( g i ( ν ) ( α ) /ν !) , v i ( t ( ν ) i ( α ) /ν !) o ≥ a − ( ν − b fulfilling condition (iii).So Proposition 2.3 gives α ′ ∈ E i with g ( α ′ ) = 0 and v i ( α ′ − α ) > b . As v i ( α − β ) ≤ b for all β ∈ A i \ { α } by (b), we conclude that α ′ = β ′ for all α = β . Hence g has precisely | A i | = dr many roots in E i . As this holds for every i and g totally splits in the maximalGalois extension L of K that is contained in all E i . Moreover, as g ∈ O K [ X ], all roots of g are actually in O L .3.4. Bounding the height of the roots of g . From condition (4) of Section 3.2, forevery coefficient a of g and every σ ∈ Hom( K, C ), we have | σ ( a ) | ≤ B := δ K N ( p ) / [ K : Q ] · n Y i =1 N ( p i ) m i / [ K : Q ] . By Proposition 2.4, g has a root α ∈ O L with h ( α ) bounded by(3.4) log( B √ deg g + 1)deg g ≤ log (cid:0) δ K N ( p ) / [ K : Q ] √ deg g + 1 (cid:1) deg g + n X i =1 m i deg g · log( q i )[ K : Q ] . By the definition of m i in (3.2) and recalling deg g = dr from (1), we have m i deg g = 1 e i · q f i i − · q f i k i i − r + k i e i r + 2 ce i r . (3.5)Condition (i) of Section 3.2 implies that q f i k i i − r ≤ ǫ. oreover, k i e i r = log( q f i k i i ) e i rf i log( q i ) ≤ de i f i log( q i ) · log(deg g )deg g ≤ C n log(deg g )deg g Finally, 2 ce i r ≤ dC n +1 e i deg g ≤ C n +1 deg g . Therefore, substituting in (3.5), and recalling that C ≥ ρ ≥
3, we have m i deg g ≤ e i ( q f i i −
1) + ǫe i ( q f i i −
1) + 11 C n +1 log(deg g )deg g . Thus the second summand in (3.4) can be bounded as n X i =1 m i deg g · log( q i )[ K : Q ] ≤ n X i =1 f ( p i | p i )[ K : Q ] · log( p i ) e i ( q f i i −
1) + nǫ + 11 nC n +2 log(deg g )deg g . As for the first summand in (3.4), note that N ( p ) / [ K : Q ] ≤ p where p is the smallestprime number not in the set { p , . . . , p n } , which, by Bertrand’s postulate, can be boundedby p < i p i ≤ C . Moreover, δ K = [ K : Q ] [ K : Q ]([ K : Q ] − p | ∆ K | ≤ C · C ( C − and thus, as C ≥ (cid:0) δ K N ( p ) / [ K : Q ] √ deg g + 1 (cid:1) deg g ≤ log (cid:16) C C − C +22 √ deg g + 1 (cid:17) deg g ≤ nC n +1 log(deg g )deg g . Therefore, from (3.4) we get h ( α ) ≤ n X i =1 f ( p i | p i )[ K : Q ] · log( p i ) e i ( q f i i −
1) + nǫ + 13 nC n +2 log(deg g )deg g , so α satisfies the height bound (1.1) of Theorem 1.3 (recalling that ǫ = 0 if n = 1). Acknowledgments
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S. Checcoli, Institut Fourier, Universit´e Grenoble Alpes, 100 rue des Math´ematiques,38610 Gi`eres, France
Email address : [email protected] A. Fehm, Institut f¨ur Algebra, Fakult¨at Mathematik, Technische Universit¨at Dres-den, 01062 Dresden, Germany
Email address : [email protected]@tu-dresden.de