On monogenity of certain pure number fields defined by x p r −m
aa r X i v : . [ m a t h . N T ] F e b ON MONOGENITY OF CERTAIN PURE NUMBER FIELDS DEFINED BY x p r − m HAMID BEN YAKKOU AND LHOUSSAIN EL FADILA bstract . Let K = Q ( α ) be a pure number field generated by a complex root α amonic irreducible polynomial F ( x ) = x p r − m , with m , p is a rational prime integer, and r is a positive integer. In this paper, westudy the monogenity of K . We prove that if ν p ( m p − m ) =
1, then K is monogenic.But if r ≥ p and ν p ( m p − m ) > p , then K is not monogenic. Some illustrating examplesare given.
1. I ntroduction
Let K be a number field defined by a monic irreducible polynomial f ( x ) ∈ Z [ x ]and Z K its ring of integers. It is well know that the ring Z K is a free Z -moduleof rank n = [ K : Q ]. Thus the abelian group Z K / Z [ α ] is finite. Its cardinal orderis called the index of Z [ α ], and denoted ( Z K : Z [ α ]). Kummer showed that for arational prime integer, if p does not divide ( Z K : Z [ α ]), then the factorization of p Z K can be derived directly from the factorization of F ( x ) in F p [ x ]. In 1878, Dedekindgave a criterion to tests wither p divides or not ( Z K : Z [ α ]) (see [5, Theorem 6.1.4]and [23]). The index i ( K ) of the field K is i ( K ) = gcd { ind ( θ ) | θ ∈ Z K generates K } . Arational prime integer p dividing i ( K ) is called a prime common index divisor of K .The ring Z K is said to have a power integral basis if it has a Z -basis (1 , θ, · · · , θ n − )for some θ ∈ Z K . That means Z K = Z [ θ ] or Z K is mono-generated as a ring,with a single generator θ . In such a case, the field K is said to be monogenicand not monogenic otherwise. The problem of checking the monogenity of thefield K is called a problem of Hasse [12, 14, 18, 19, 20]. The problem of testingthe monogenity of number fields and constructing power integral bases have beenintensively studied, this last century, mainly by Ga´al, Nakahara, Peth ¨o, and theirresearch groups (see for instance [3, 13, 15, 22]). In [6], El Fadil gave conditionsfor the existence of power integral bases of pure cubic fields in terms of the indexform equation. In [12], Funakura, studied the integral basis in pure quartic fields.In [16], Ga´al and Remete, calculated the elements of index 1 (with coe ffi cients withabsolute value < in the integral basis) of pure quartic fields generated by m for 1 < m < and m ≡ , m ≡ , m . ∓ m is monogenic. They also showed in [3], that if m ≡ m . ∓ m is not monogenic. Also, Hameed and Date : February 9, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Power integral basis, Theorem of Ore, prime ideal factorization.
Nakahara [4], proved that if m ≡ m / is not monogenic, but if m ≡ , m n , where 3 ≤ n ≤
9. While Ga´al’sand Peth ¨o’s techniques are based on the index calculation, Nakahara’s methods arebased on the existence of power relative integral bases of some special sub-fields.The goal of this paper is to study the monogenity of pure number fields of degree p n for every rational prime integer p and n is a natural integer. Our proposed resultsgeneralize [2, Theorem 2.1], where pure number fields of degree 2 n are previouslystudied. Our method is based on prime ideal factorization and is similar to that usedin [7, 8, 9]. 2. M ain results Let K be the number field defined by a complex root α of a monic irreduciblepolynomial F ( x ) = x p r − m , with m , Theorem 2.1. If ν p ( m p − m ) = , then Z [ α ] is the ring of integers of K; K is monogenic and α generates a power integral basis. Corollary 2.2.
For p = , if m ≡ or , then Z [ α ] is integrally closed for every naturalinteger r. So that Theorem 2.1 generalizes [2, Theorem 2.1] , where p = is previouslyconsidered. Theorem 2.3.
Assume that p is an odd prime integer and p does not divide m. If ν p ( m p − − > p and r ≥ p, then K is not monogenic. Corollary 2.4.
Assume that p = . (1) If m . ∓ , then K is monogenic. (2) If r ≥ and m ≡ ∓ , then K is not monogenic. Theorem 2.5.
Assume that p = . (1) If r = and m ≡ , then the pure quartic number field K is not monogenic. (2) If r ≥ and m ≡ , then K is not monogenic.
3. P roofs
We start by recalling some fundamental notions on Newton polygon’s techniques.For more details, we refer to [10, 11]. For any prime integer p and for any monicpolynomial φ ∈ Z [ x ] whose reduction is irreducible in F p [ x ], let F φ be the field F p [ x ]( φ ) .For any monic polynomial f ( x ) ∈ Z [ x ], upon the euclidean division by successivepowers of φ , we expand f ( x ) as f ( x ) = P li = a i ( x ) φ ( x ) i , called the φ -expansion of f ( x )(for every i , deg ( a i ( x )) < deg ( φ )). The φ -Newton polygon of f ( x ) with respect to p , isthe lower boundary convex envelope of the set of points { ( i , ν p ( a i ( x ))) , a i ( x ) , } in theeuclidean plane, which we denote by N φ ( f ). For every i , j = , . . . , l , let a i = a i ( x ) and µ ij = ν p ( a i ) − ν p ( a j ) i − j ∈ Q . Then we obtain the following integers 0 = i < i < · · · < i r = l satisfying i j + = max { i = i j + , . . . l , µ i j i j + ≤ µ i j i } . For every j = , . . . r , let S j be the segment joining the points A j − = ( i j − , ν ( a i j − )) and A j = ( i j , ν ( a i j )) in the euclideanplane. The segments S , . . . , S r are called the sides of the polygon N φ ( f ). For every j = , . . . , r , the rational number λ j = ν p ( a ij ) − ν p ( a ij − ) i j − i j − ∈ Q is called the slope of S j , l ( S j ) = i j − i j − is its length, and h ( S j ) = − λ j l ( S j ) is its height. In what follows ν ( a i j ) = ν ( a i j − ) + l ( S j ) λ j . The φ -Newton polygon of f , is the process of joining thesegments S , . . . , S r ordered by the increasing slopes, which can be expressed as N φ ( f ) = S + · · · + S r . Notice that N φ ( f ) = S + · · · + S r is only a notation and not thesum in the euclidean plane. For every side S of the polygon N φ ( f ), l ( S ) is the lengthof its projection to the x -axis and h ( S ) is the length of its projection to the y -axis. Theprincipal part of N φ ( f ), denoted N + φ ( f ), is the part of the polygon N φ ( f ), which isdetermined by joining all sides of negative slopes.For every side S of N φ ( f ), with initial point ( s , u s ) and length l , and for every i = , . . . , l , we attach the following residual coe ffi cient c i ∈ F φ as follows: c i = , if ( s + i , u s + i ) lies strictly above S , a s + i ( x ) p u s + i ! (mod ( p , φ ( x ))) , if ( s + i , u s + i ) lies on S . where ( p , φ ( x )) is the maximal ideal of Z [ x ] generated by p and φ . That means if( s + i , u s + i ) lies on S , then c i = a s + i ( β ) p u s + i , where β is a root of φ .Let λ = − h / e be the slope of S , where h and e are two positive coprime integers, andlet d = l / e be the degree of S . Notice that, the points with integer coordinates lying in S are exactly ( s , u s ) , ( s + e , u s + h ) , · · · , ( s + de , u s + dh ). Thus, if i is not a multiple of e , then( s + i , u s + i ) does not lie in S , and so, c i =
0. Let f S ( y ) = t y d + t y d − + · · · + t d − y + t d ∈ F φ [ y ],called the residual polynomial of f ( x ) associated to the side S , where for every i = , . . . , d , t i = c ie . Remark.
Note that if ν ( a s ( x )) = λ =
0, and φ = x , then F φ = F p and for every i = , . . . , l , c i = a s + i (mod p ). Thus this notion of residual coe ffi cient generalizes thereduction modulo a maximal ideal and f S ( y ) ∈ F p [ y ] coincides with the reduction of f ( x ) modulo the maximal ideal ( p ).Let N + φ ( f ) = S + · · · + S r be the principal φ -Newton polygon of f with respect to p .We say that f is a φ -regular polynomial with respect to p , if for every i = , . . . , r , f S i ( y )is square free in F φ [ y ]. We say that f is a p -regular polynomial if f is a φ i -regularpolynomial with respect to p for every i = , . . . , t , for some monic polynomials φ , . . . , φ t with f ( x ) = Q ti = φ il i is the factorization of f ( x ) in F p [ x ].The theorem of Ore plays a fundamental key for proving our main Theorems:Let φ ∈ Z [ x ] be a monic polynomial, with φ ( x ) is irreducible in F p [ x ]. As defined in[11, Def. 1.3], the φ -index of f ( x ), denoted by ind φ ( f ), is deg( φ ) times the numberof points with natural integer coordinates that lie below or on the polygon N + φ ( f ),strictly above the horizontal axis, and strictly beyond the vertical axis (see Figure 1). HAMID BEN YAKKOU AND LHOUSSAIN EL FADIL S S S F igure N + φ ( f ). Now assume that f ( x ) = Q ti = φ il i is the factorization of f ( x ) in F p [ x ], where every φ i ∈ Z [ x ] is monic, φ i ( x ) is irreducible in F p [ x ], φ i ( x ) and φ j ( x ) are coprime when i , j = , . . . , t . For every i = , . . . , t , let N + φ i ( f ) = S i + · · · + S ir i be the principalpart of the φ i -Newton polygon of f with respect to p . For every j = , . . . , r i , let f S ij ( y ) = Q s ij s = ψ a ijs ijk ( y ) be the factorization of f S ij ( y ) in F φ i [ y ]. Then we have thefollowing theorem of Ore (see [11, Theorem 1.7 and Theorem 1.9], [10, Theorem 3.9],and [21]): Theorem 3.1. (Theorem of Ore) (1) ν p ( ind ( f )) = ν p (( Z K : Z [ α ])) ≥ P ti = ind φ i ( f ) and equality holds if f ( x ) is p-regular;every a i , j , s = . (2) If every a i , j , s = , then p Z K = r Y i = r i Y j = s ij Y s = p e ij i , j , s , where e ij is the ramification index of the side S ij and f ijs = deg ( φ i ) × deg ( ψ ijs ) is theresidue degree of p ijs over p. Corollary 3.2.
Under the hypothesis of Theorem 3.1, if for every i = , . . . , t, l i = orN φ i ( f ) = S i has a single side of height , then ν p ( ind ( f )) = . The following lemma allows to evaluate the p -adic valuation of the binomialcoe ffi cient (cid:0) p r j (cid:1) . Lemma 3.3.
Let p be a rational prime integer and r be a positive integer. Then ν p (cid:16)(cid:0) p r j (cid:1)(cid:17) = r − ν p ( j ) for any integer j = , . . . , p r − .Proof. Since for any natural number m , ν p ( m !) = P mt = ν p ( t ), we have ν p (cid:16)(cid:0) p r j (cid:1)(cid:17) = ν p ( p r !) − ν p (( p r − j )!) − ν p ( j !) = P p r t = ν p ( t ) − P p r − jt = ν p ( t ) − P jt = ν p ( t ). Thus ν p (cid:16)(cid:0) p r j (cid:1)(cid:17) = P p r t = p r − j + ν p ( t ) − P jt = ν p ( t ) = ν p ( p r ) + P j − t = ν p ( p r − t ) − P jt = ν p ( t ). As for every t = , . . . , j − < p r , then ν p ( p r − t ) = ν p ( t ). Hence ν p (cid:16)(cid:0) p r j (cid:1)(cid:17) = r − ν p ( j ). (cid:3) Proof. of Theorem 2.1.Since △ ( f ) = ∓ p rp r m p r − , then by the formula linking △ ( f ), ( Z K : Z [ α ]) and d K , Z [ α ] isintegrally closed if and only if q does not divide ( Z K : Z [ α ]) for every rational primeinteger q dividing pm . Let q be a rational prime dividing m , then f ( x ) ≡ φ p r ( mod q ),where φ = x . As m is a square free integer, the φ -principal Newton polygon withrespect to ν q , N + φ ( f ) = S has a single side of degree 1. Thus f S ( y ) is irreducibleover F φ ≃ F q . By Theorem 3.1, we get ν q (( Z K : Z [ α ])) = ind φ ( f ) = q does notdivide ( Z K : Z [ α ]). Now, we deal with p , when p does not divide m . In this case, let ν = v p ( m p − − f ( x ) = ( x − m + m ) p r − m = ( x − m ) p r + P p r − j = (cid:0) p r j (cid:1) m p r − j ( x − m ) j + m p r − m ,then by Lemma 3.3, f ( x ) ≡ φ p r ( mod p ), where φ = x − m . If ν =
1, then N + φ ( f ) = S hasa single side of height 1 and by Theorem 3.1, p does not divide ( Z K : Z [ α ]). But if ν ≥
2, then p divides ( Z K : Z [ α ]). (cid:3) For the proof of Theorem 2.3, Corollary 2.4, and Theorem 2.5, we need the follow-ing Lemma, which is an immediate consequence of a theorem of Kummer.
Lemma 3.4.
Let p be rational prime integer and K be a number field. For every positiveinteger f , let P f be the number of distinct prime ideals of Z K lying above p with residuedegree f and N f be the number of monic irreducible polynomials of F p [ x ] of degree f . IfP f > N f , then p is a common index divisor of K.Proof. of Theorem 2.3.Under the hypothesis of Theorem 2.3, F ( x ) = ( x − m ) p r in F p [ x ] and F ( x ) = ( x − m + m ) p r − m = ( x − m ) p r + P p r − j = (cid:0) p r j (cid:1) m p r − j ( x − m ) j + m p r − m . Let φ = x − m and ν = ν p ( m p r − m ).As m p r − m = ( m p − m )( m t + · · · + t = p r −
1, if m ≡ p ν ), then m t + · · · + ≡− p ), and so, ν p ( m t + · · · + =
0. Thus ν p ( m p − − = ν p ( m p − m ) = ν p ( m p r − m ).It follows that if ν > p and r ≥ p , then by Lemma 3.3, N + φ ( f ) = S + · · · + S t − p + + · · · + S t has t -distinct sides of degree 1 each one, with t ≥ p +
1. More precisely, S t − i has( p r − i − , i +
1) as the first point and ( p r − i , i ) as the last point for every i = , . . . , p − S has (0 , ν ) as the first point and (1 , r ) or ( p r − ν + , ν −
1) as the last point accordingto ν > r or r ≥ ν . It follows by Theorem 3.1 that, there are t distinct prime ideals of Z K lying above p ,with residue degree 1 each one. As t = P > p = N , by Lemma3.4, p is a common index divisor of K and K is not monogenic. (cid:3) Proof. of Theorem 2.5.Let m be an odd rational integer. Then F ( x ) = ( x − p r in F p [ x ] and F ( x ) = ( x − + p r − m = ( x − p r + P p r − j = (cid:0) p r j (cid:1) ( x − j + − m . Let φ = x − ν = ν p (1 − m ). It followsthat:(1) If r = ν ≥
4, then by Lemma 3.3, N + φ ( f ) = S + S + S has 3-distinct sides,with degree 1 each side. Thus there are 3-distinct prime ideals of Z K lyingabove 2 with residue degree 1 each one. So, by Lemma 3.4, 2 is a commonindex divisor of K .(2) If r ≥ ν ≥
5, then by Lemma 3.3, N + φ ( f ) = S + S + · · · + S t has t -distinctsides, with t ≥ S i equals 1 for every i = , . . . , t . Thus there are HAMID BEN YAKKOU AND LHOUSSAIN EL FADIL at least t − Z K lying above 2 with residue degree 1each one. As t − ≥
3, then by Lemma 3.4, 2 is a common index divisor of K . (cid:3) Examples. (1) If m < { , , , , , } (mod 49), then the pure number field Q ( r √ m ) is monogenic for every positive integer r .(2) If m p − ≡ p p + ), then for every natural integer r ≥ p the pure field Q ( α )is not monogenic. In particular, if m ≡ ), then the pure field Q ( r √ m )is not monogenic, for every natural integer r ≥ m ≡ Q ( √ m ) is not monogenic .(4) More general, if m ≡ r ≥ Q ( α ) isnot monogenic, where α is a complex root of x r − m .(5) If ν ( m − ≥
4, the field Q ( √ m ) is not monogenic. ( see figure 2 ).(6) More general, if m ≡ ∓ Q ( α ) is not monogenic, where α is a complex root of x r − m and r ≥ v S S S S F igure The ( x − m )-Newton polygon for F ( x ) = x − m at p = v ≥ R eferences [1] S. A hmad , T. N akahara , and S. M. H usnine , Power integral bases for certain pure sextic fields , Int.J. of Number Theory v:10, No 8 (2014) 2257– 2265.[2] A. Hameed, T. Nakahara, S. M. Husnine, On existence of canonical number system in certainclasses of pure algebraic number fields, J. Prime Res. Math. 7(2011) 19-24.[3] S. A hmad , T. N akahara , and A. H ameed , On certain pure sextic fields related to a problem of Hasse ,Int. J. Alg. and Comput. 26(3) (2016) 577–583[4] A.H ameed and
T.N akahara , Integral bases and relative monogenity of pure octic fields , Bull. Math.Soc. Sci. Math. R ´epub. Soc. Roum. 58(106) No. 4(2015) 419–433[5] H. Cohen,
A Course in Computational Algebraic Number Theory , GTM 138, Springer-Verlag BerlinHeidelberg (1993)[6] L. El Fadil,
Computation of a power integral basis of a pure cubic number field , Int. J. Contemp. Math.Sci. 2(13-16)(2007) 601–606[7] L. El Fadil,
On Power integral bases for certain pure sextic fields (To appear in the forthcoming issueof Bol. Soc. Paran. Math.)[8] L. El Fadil,
On Power integral bases for certain pure number fields (preprint) [9] L. El Fadil,
On Power integral bases for certain pure number fields defined by x − m (To appear inthe forthcoming issue of Stud. Sci. Math. Hung.)[10] L. El Fadil, On Newton polygon’s techniques and factorization of polynomial over henselian valued fields ,J. of Algebra and its Appl. (2020), doi: S0219498820501881[11] L. E l F adil , J. M ontes and E. N art , Newton polygons and p-integral bases of quartic number fields , J.Algebra and Appl. 11(4)(2012) 1250073[12] T. F unakura , On integral bases of pure quartic fields , Math. J. Okayama Univ. 26 (1984) 27-–41[13] I. G a ´ al , Power integral bases in algebraic number fields , Ann. Univ. Sci. Budapest. Sect. Comp. 18(1999) 61–87[14] I. G a ´ al , Diophantine equations and power integral bases, Theory and algorithm, Second edition,Boston, Birkh¨auser, 2019[15] I. G a ´ al , P. O lajos , and M. P ohst , Power integral bases in orders of composite fields , Exp. Math. 11(1)(2002) 87–90.[16] I. G a ´ al and L. R emete , Binomial Thue equations and power integral bases in pure quartic fields , JPJournal of Algebra Number Theory Appl. 32(1) (2014) 49–61[17] I. G a ´ al and L. R emete , Power integral bases and monogenity of pure fields , J. of Number Theory 173(2017) 129–146[18] H. H asse , Zahlentheorie , Akademie-Verlag, Berlin, 1963.[19] K. H ensel , Theorie der algebraischen Zahlen , Teubner Verlag, LeipzigBerlin, 1908.[20] Y. Motoda, T. Nakahara and S. I. A. Shah,
On a problem of Hasse , J. Number Theory 96 (2002)326—334[21] O. O re , Newtonsche Polygone in der Theorie der algebraischen Korper , Math. Ann., 99 (1928), 84–117[22] A. P eth ¨ o and M. P ohst , On the indices of multiquadratic number fields , Acta Arith. 153(4) (2012)393–414[23] R. Dedekind, ¨Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der h¨oherenKongruenzen , G ¨ottingen Abhandlungen (1878) 1–23F aculty of S ciences D har E l M ahraz , P.O. B ox tlas -F ez , S idi M ohamed B en A bdellah U niversity , F ez – M orocco Email address ::