EXPLICIT DECOMPOSITION OF A RATIONAL PRIME IN A CUBIC FIELD

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1 EXPLICIT DECOMPOSITION OF A RATIONAL PRIME IN A CUBIC FIELD ŞABAN ALACA, BLAIR K. SPEARMAN, AND KENNETH S. WILLIAMS Received 19 June 5; Revised 19 February 6; Accepted 1 March 6 We give the explicit decomposition of the principal ideal p (p prime) in a cubic field. Copyright 6 Hindawi Publishing Corporation. All rights reserved. 1. Introduction Let K be an algebraic number field. Let O K denote the ring of integers of K.Letd(K)denote the discriminant of K.Letθ O K be such that K = Q(θ). The minimal polynomial of θ over Q is denoted by irr Q (θ). The discriminant D(θ) and the index ind(θ) ofθ are related by the equation D(θ) = ( ind(θ) ) d(k). (1.1) If p is a prime not dividing ind(θ), then it is well known that the following theorem of Dedekind gives explicitly the factorization of the principal ideal p of O K into prime ideals in terms of the irreducible factors of irr Q (θ)modulop;see,forexample,[,theorem 1.5.1, page 57]. Theorem 1.1. Let K = Q(θ) be an algebraic number field with θ O K.Letp be a rational prime. Let f (x) = irr Q (θ) Z[x]. (1.) Let denote the natural map Z[x] Z p [x],wherez p = Z/pZ.Let f (x) = g 1 (x) e1 g r (x) er, (1.) where g 1 (x),...,g r (x) are distinct irreducible polynomials in Z p [x],ande 1,...,e r are positive Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 6, Article ID 17641, Pages 1 11 DOI /IJMMS/6/17641

2 Decomposition of primes in a cubic field integers. For i = 1,,...,r,let f i (x) be any polynomial of Z[x] such that f i = g i and deg( f i ) = deg(g i ).Set P i = p, f i (θ), i = 1,,...,r. (1.4) If ind(θ) (modp), then P 1,...,P r are distinct prime ideals of O K with p =P1 e1 Pr er, N ( ) P i = p deg f i, i = 1,,...,r. (1.5) On the other hand if p is a prime dividing ind(θ), no such general theorem is known which gives the prime ideals explicitly, and all that is available in general is the Buchmann- Lenstra algorithm [4, page 15] for decomposing a prime in a number field. If p is not a common index divisor of K, then there exist elements φ O K for which K = Q(φ), and p ind(φ), and we can apply Dedekind s theorem to obtain the prime ideal factorization of p from the minimal polynomial irr Q (φ). However given θ it is not easy to determine such an element φ in general. Moreover when p is a common index divisor of K,nosuch element φ exists and Dedekind s theorem cannot be applied. In this paper we treat the case when K is a cubic field and p is a prime dividing ind(θ). When p is a common index divisor of K (the only possibility is p = ), we quote the results in []. When p is not a common index divisor, we construct an element φ O K such that K = Q(φ) andp ind(φ) and then apply Dedekind s theorem to obtain the prime ideal factorization of p. Our construction of φ was guided by the p-integral bases of K given by Alaca [1]. We give explicitly the prime ideals in the factorization of p into prime ideals in O K. The form of the prime ideal factorization has been given by Llorente and Nart [6, Theorem 1, page 58] and we make use of their results. A method for factoring all primes in a cubic field is given in [5, pages ]. It is well known that K can be given in the form K = Q(θ), where θ is a root of the irreducible polynomial f (x) = x ax + b, a,b Z, (1.6) so that irr Q (θ) = f (x). Moreover it is further known that a and b can be chosen so that there are no primes p with p a and p b.wehave D(θ) = 4a 7b. (1.7) Let ν p (k) denote the largest nonnegative integer m such that p m divides the nonzero integer k.from(1.1)wededucethat ( ) ( ) ( ) ν p D(θ) νp d(k) ν p ind(θ) =. (1.8) We set D p (θ) = D(θ) ( ). (1.9) p νp D(θ)

3 ŞabanAlacaetal. Table 1.1. p =. Case Conditions ν (d(k)) ν (ind(θ)) φ A1 A A A4 A5 a (4), b 4(8) ν (D(θ)) = 4 a (4), b (8) ν (D(θ)) = 5 a (4), b 4(8) ν (D(θ)) = 4 a 1(4), b (4) D (θ) 1(8) ν (D(θ)) = a 1(4), b (4) D (θ) 5(8) ν (D(θ)) = 1 θ 1 1+θ + θ 1 θ 1 θ + θ 1 θ + θ Factors of Prime ideals Norms P P =,φ N(P) = PQ P =,1 + φ Q =,φ PQ PQR PQ P =,φ Q =,1 + φ P =,θ Q =,1 + φ R =,1 + θ + φ N(P) = N(Q) = N(P) = N(Q) = N(P) = N(Q) = N(R) = P =,φ Q =,1 + φ + φ N(P) = N(Q) = 4 A6 a (4), b (4) ν (D(θ)) 1() ν (D(θ)) 5 ν (D(θ)) 1+λ + λ λ = m+1 m = ν (D(θ)) PQ P =,1 + φ Q =,φ N(P) = N(Q) = A7 a (4), b (4) ν (D(θ)) () ν (D(θ)) 4 D (θ) (4) ν (D(θ)) m+1 m = ν (D(θ)) PQ P =,φ Q =,1 + φ N(P) = N(Q) = A8 a (4), b (4) ν (D(θ)) () ν (D(θ)) 4 D (θ) 1(8) ν (D(θ)) m m = ν (D(θ)) PQR P =,φ +φ + φ Q =, +φ + φ R =, N(P) = N(Q) = N(R) = A9 a (4), b (4) ν (D(θ)) () ν (D(θ)) 4 D (θ) 5(8) ν (D(θ)) λ + λ λ = m+1 m = ν (D(θ)) PQ P =,φ Q =,1 + φ + φ N(P) = N(Q) = 4 The determination of ν p (d(k)) was carried out by Llorente and Nart [6,Theorem,page 58] in 198; see also Alaca [1]. The values of ν p (D(θ)) and ν p (d(k)) are listed in tabular form in Alaca [1] depending on congruence conditions on a and b.from[1]wededuce that p ind(θ) in precisely those cases listed in Tables 1.1, 1., 1., and no others. We abbreviate r s (modm)byr s(m). In the sixth column of each table we give the form of the prime ideal factorization from the work of Llorente and Nart [6,Theorem1,page 58]. However, Llorente and Nart did not give the prime ideals explicitly. We give explicit formulae for these prime ideals in the seventh column of each of Tables 1.1, 1.,and1.. It is convenient to set = 4a +9bθ +6aθ O K. (1.1)

4 4 Decomposition of primes in a cubic field Case Conditions ν (d(k)) ν (ind(θ)) φ Table 1.. p =. Factors of Prime ideals Norms B1 = ν (b) = ν (a) ν (D(θ)) = θ P P =,φ N(P) = B B B4 = ν (b) < ν (a) ν (D(θ)) = 7 1 = ν (a) < ν (b) ν (D(θ)) = ν (a) 1, ν (b) = a (9), b a +1(9) ν (D(θ)) = 5 1 θ θ + θ, ( ), θ + θ,( ). ( )ifa b (7) ( )ifa + b (7) see Note 1 bθ + θ, ( ), 1+bθ + θ,( ). ( )if9 a +1 b ( )if7 a +1 b P P =,φ N(P) = PQ PQ P =,φ a N(P) = Q =,φ N(Q) = (a +) P =, Q =,φ, ( ) a P =, + φ Q =,1 + φ, ( ) + φ N(P) = N(Q) = B5 B6 a (9), ν (b) = b 4(9), b a + 1(7) ν (D(θ)) = 5 a (9), ν (b) = b a + 1(7) ν (D(θ)) 1() ν (D(θ)) ν (D(θ)) 1 1 bθ + θ λ + λ,( ) λ + λ, ( ) λ = m+ m = ν (D(θ)) ( )ifa m 1 D (θ)(9) ( )ifa m 1 D (θ)(9) see Note P P =,φ N(P) = PQ P =,φ Q =,φ ad (θ) N(P) = N(Q) = B7 a (9), ν (b) = b a + 1(7) ν (D(θ)) () ν (D(θ)) 6 D (θ) () ν (D(θ)) m+ m = ν (D(θ)) PQ P =, + φ Q =, + φ + φ if m =, P =,φ Q =,1 + φ if m N(P) = N(Q) = 9 B8 a (9), ν (b) = b a + 1(7) ν (D(θ)) () ν (D(θ)) = 6 D (θ) 1() P P = N(P)= 7

5 ŞabanAlacaetal. 5 Table 1.. Continued. Case Conditions ν (d(k)) ν (ind(θ)) φ B9 a (9), ν (b) = b a + 1(7) ν (D(θ)) () ν (D(θ)) 8 D (θ) 1() ν (D(θ)) m+ m = ν (D(θ)) Note: In case B (resp., B6) if b (7) (resp., m ), both choices for φ are valid. Factors of PQR Prime ideals P =,φ Q =, 1+φ R =,1 + φ Norms N(P) = N(Q) = N(R) = Table 1.. p>. Case Conditions ν p (d(k)) ν p (ind(θ)) φ C1 C = ν p (b) ν p (a) ν p (D(θ)) = 4 1 = ν p (a) < ν p (b) ν p (D(θ)) = Factors of p Prime ideals Norms 1 θ P p P = p,φ N(P) = p θ p, if p b, P = p,φ 1 1 PQ θ + θ Q = p, a N(P) = p p,ifp b. p + φ N(Q) = p C C4 ν p (a) = ν p (b) = ν p (D(θ)) 1() ν p (D(θ)) ν p (a) = ν p (b) = ν p (D(θ)) () ν p (D(θ)) ( ) Dp (θ) = 1 p 1 ν p (D(θ)) 1 ν p (D(θ)) λ + λ p λ = p m m = ν p(d(θ)) 1 p m m = ν p(d(θ)) t ad p (θ)(p) PQ PQR P = p,φ Q = p, ad p (θ)+φ P = p,φ Q = p, t + φ R = p,t + φ N(P) = p N(Q) = p N(P) = p N(Q) = p N(R) = p C5 ν p (a) = ν p (b) = ν p (D(θ)) () ν p (D(θ)) ( ) Dp (θ) = 1 p ν p (D(θ)) p m m = ν p(d(θ)) PQ P = p,φ Q = p, ad p (θ)+φ N(P) = p N(Q) = p It is easy to show that the minimal polynomial of over Q is q(x) = x ad(θ)x + D(θ) (1.11) and that disc ( q(x) ) = 6 b D(θ). (1.1)

6 6 Decomposition of primes in a cubic field. Case A1 In this case we can define integers A and B by a = 4A and b = 8B +4.Set θ /. The minimal polynomial of φ over Q is so that φ O K.Further p(x) = x 4Ax +4A x ( 8B +8B + ) (.1) disc ( p(x) ) = 4(B +1) ( 18B + 18B 16A +7 ). (.) We have p(x) x (mod). As disc ( p(x) ), d(k), we have ind(φ), so that by Theorem 1.1,. Cases A, A, A5, A7, B1, B, B5, B7, B9, C1, C These cases can be treated similarly to case A1. 4. Cases A4, A8 =,φ. (.) In these cases is a common index divisor and we can appeal to [6, Theorem 4, page 585] for the results. 5. Case A6 We let λ = / m+1,whereν (D(θ)) = m + 5, and 1+λ + λ /. By (1.11), the minimal polynomial of over Q is x ad(θ)x + D(θ) so that the minimal polynomial of λ over Q is x ad(θ) D(θ) x + m+ m+ = x 6aD (θ)x + m+ D (θ). (5.1) Hence λ O K.WearenowincaseAwith a = 6aD (θ) (mod4), b = m+ D (θ) (mod8), D(θ) = 6 b D(θ) 6m+6, ( ) ν D(θ) = +(m +) (6m +6)= 5. (5.) Thus by case A we obtain =,φ +1,φ. (5.)

7 6. Case A9 ŞabanAlacaetal. 7 In this case we set ν (D(θ)) = m + (so that m 1), λ = / m+1,and (λ + λ )/. Then proceeding as in case A6 we can reduce this case to case A5. 7. Case B In this case we have 1 = ν (a) < ν (b), ν ( D(θ) ) =. (7.1) Clearly 9 a b and 9 a + b. However 7 cannot divide both of a b and a + b as their sum 6a is not divisible by 7. Hence we can define θ + θ if a b (mod7), θ θ if a + b (mod7). (7.) We note that if 7 b we can choose either value of θ / ± θ for φ.set +1 if a b (mod7), ε = 1 ifa + b (mod7), (7.) subjecttotheremarkabove,sothat θ + εθ. (7.4) The minimal polynomial of φ is p(x) = x a ( ) x + a + a 9 + εb x + εb b 7 εab 9 (7.5) so that φ O K.Wehave p(x) x a x + (x a 9 x x a ) (mod). (7.6) Further disc ( p(x) ) D(θ)(a εb 7) = 6. (7.7) As disc(p(x)), d(k), we have ind(φ), so that by Theorem 1.1,weobtain =,φ,φ a. (7.8)

8 8 Decomposition of primes in a cubic field 8. Case B4 In this case we have 9 a +1 b.weset ( θ bθ +1 ) ( θ +bθ +1 ) if 9 a +1 b, if 7 a +1 b. Firstweconsiderthecase9 a +1 b. The minimal polynomial of φ is (8.1) p(x) = x (a +) so that p(x) Z[x]andφ O K.Wehave x + (a +)( a +1 b ) ( a +1 b ) x 9 7 p(x) x ( x (8.) ) a + (mod). (8.) Further disc ( p(x) ) ( a +1 b = b ) D(θ) 6 (8.4) so that disc(p(x)), d(k), thus ind(φ), and by Theorem 1.1 we have =,φ,φ a +. (8.5) Now we turn to the case 7 a +1 b. The minimal polynomial of φ is p(x) = x + p x + p 1 x + p, (8.6) where Clearly (a +) p =, ( a p +4a 4ab +6b + ) 1 =, 9 ( a p a +ab +8b 4 7b 1 ) =. 7 ( a +1 b p 1 = (1a 18) 7 ( a(a +1) a +1 b p = +9 7 p Z, )( 4 ) ( ) a ( ) a + +1 Z, ( a +1 b 7 ) ) (a +1) Z, (8.7) (8.8)

9 ŞabanAlacaetal. 9 so that φ O K.Further p a +(mod), p 1 a +1(mod), (8.9) p a (mod). Hence ( p(x) x + a ) (x +1) (mod). (8.1) Further As a,6 (mod9), a +1 b (mod7), and we see that disc ( p(x) ) ( 8b = b D(θ) a +1 ) 6. (8.11) 8b a +1= 6(a ) 8 ( a +1 b ) + 7; (8.1) 8b a + 1 (8.1) so that disc(p(x)), d(k), and thus ind(φ). Hence by Theorem 1.1 we have =,φ + a,φ +1. (8.14) 9. Case B6 In this case we set ν (D(θ)) = m +sothatm. Let λ = m+, λ + λ if a m 1 D (θ)(mod9), λ λ if a m 1 D (θ)(mod9). (9.1) The minimal polynomial of λ is p(x) = x ad (θ)x + m D (θ), disc ( p(x) ) = b D (θ). (9.)

10 1 Decomposition of primes in a cubic field We are now in case B with a = ad (θ), ν (a ) = 1, b = m D (θ) (mod9), 4a 7b = b D (θ), ν ( 4a 7b ) =. (9.) Hence =,φ,φ ad (θ). (9.4) 1. Case B8 Here is a prime ideal. 11. Case C Similarly to case B6 this case can be reduced to case C. 1. Cases C4, C5 Here ( ) ( ) p a, p b, ν p D(θ) (mod), νp D(θ), ( ) Dp (θ) +1, case C4, = p 1, case C5. (1.1) Set ν p (D(θ)) = m so that m 1. Let /p m. The minimal polynomial of φ is p(x) = x ad p (θ)x + p m D p (θ), disc ( p(x) ) = 6 b D(θ) p 6m. (1.) Clearly p disc(p(x)) so that p ind(φ). Now p(x) x ( x ad p (θ) ) (mod p). (1.) As 4a 7b (modp), p a, p b, p>, (1.4) we have ( ) a = 1. (1.5) p

11 Şaban Alacaetal. 11 Thus (x x t)(x + t)(modp), case C4, ad p (θ) irreducible (mod p), case C5, (1.6) where t ad p (θ)(modp). Hence p,φ p,φ t p,φ + t, p = p,φ p,φ ad p (θ), where N( p,φ ad p (θ) ) = p. casec4, casec5, (1.7) Acknowledgment The research of the second and third authors was supported by grants from the Natural Sciences and Engineering Research Council of Canada. References [1] Ş.Alaca, p-integral bases of a cubic field, Proceedings of the American Mathematical Society 16 (1998), no. 7, [] Ş. Alaca, B. K. Spearman, and K. S. Williams, The factorization of in cubic fields with index, Far East Journal of Mathematical Sciences (FJMS) 14 (4), no., 7 8. [] Ş. Alaca and K.S. Williams, Introductory Algebraic Number Theory, Cambridge University Press, Cambridge, 4. [4] H. Cohen, A Course in Computational Algebraic Number Theory, Springer, New York,. [5] B. N. DeloneandD. K. Faddeev, The Theory of Irrationalities of the Third Degree, Translations of Mathematical Monographs, vol. 1, American Mathematical Society, Rhode Island, [6] P. Llorente and E. Nart, Effective determination of the decomposition of the rational primes in a cubic field, Proceedings of the American Mathematical Society 87 (198), no. 4, Şaban Alaca: Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University Ottawa, ON, Canada K1S 5B6 address: salaca@math.carleton.ca Blair K. Spearman: Department of Mathematics and Statistics, University of British Columbia Okanagan, Kelowna, BC, Canada V1V 1V7 address: blair.spearman@ubc.ca Kenneth S. Williams: Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University Ottawa, ON, Canada K1S 5B6 address: kwilliam@connect.carleton.ca