The prime ideal decomposition of 2 in a pure quartic field with field index 2 is determined explicitly.</p

Spearman, Blair K.

Williams, Kenneth S.

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 48, Issue 1 2006 Article 5JANUARY 2006The Prime Ideal Factorization of 2 in PureQuartic Fields with Index 2Blair K. Spearman∗ Kenneth S. Williams†∗Statistics University†Statistics Carleton UniversityCopyright c©2006 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouThe Prime Ideal Factorization of 2 in PureQuartic Fields with Index 2Blair K. Spearman and Kenneth S. WilliamsAbstractThe prime ideal decomposition of 2 in a pure quartic field with field index 2 is determinedexplicitly.KEYWORDS: pure quartic field, discriminant, prime decompositionMath. J. Okayama Univ. 48 (2006), 43{46THE PRIME IDEAL FACTORIZATION OF 2 IN PUREQUARTIC FIELDS WITH INDEX 2Blair K. SPEARMAN and Kenneth S. WILLIAMSAbstract. The prime ideal decomposition of 2 in a pure quartic ¯eldwith ¯eld index 2 is determined explicitly.1. IntroductionLet K be an algebraic number ¯eld and OK its ring of integers. Whendetermining generators of the ideals in the prime ideal factorization of a(rational) prime p in OK , the most di±cult case occurs when p divides the¯eld index i(K) of K. In this paper we examine the case when K is a purequartic ¯eld. Here i(K) = 1 or 2, and we determine explicit generators ofthe prime ideals in the decomposition of 2 when i(K) = 2.Let K be a pure quartic ¯eld. Then there exists a fourth power freeinteger m such that K = Q(m1=4). It follows from the work of Funakura [1,p. 36] that the ¯eld index i(K) of K is given byi(K) =½2; if m ´ 1 (mod 16);1; if m 6´ 1 (mod 16):From now on we assume that i(K) = 2 so that m ´ 1 (mod 16), say m =16k + 1. In this case the prime ideal factorization of < 2 > in OK is< 2 >= P 21P2P3;where P1, P2, P3 are distinct prime ideals, see [1, p. 36]. In this paper wedetermine explicit generators of P1, P2 and P3.Theorem. Let m be a fourth power free integer such that K = Q(m1=4)is a pure quartic ¯eld with i(K) = 2. Then < 2 >= P 21P2P3, where theMathematics Subject Classi¯cation. 11R16.Key words and phrases. pure quartic ¯eld, discriminant, prime decomposition.Both authors were supported by research grants from the Natural Sciences and Engi-neering Research Council of Canada.431Spearman and Williams: The Prime Ideal Factorization of 2 in Pure Quartic Fields withProduced by The Berkeley Electronic Press, 200644 B. K. SPEARMAN AND K. S. WILLIAMSdistinct prime ideals P1, P2, P3 of OK are given byP1 = < 2;32+m1=4 +12m1=2 >;P2 =8>><>>:< 2;54+14m1=4 +14m1=2 +14m3=4 >; if m ´ 1 (mod 32);< 2;34+54m1=4 +34m1=2 +14m3=4 >; if m ´ 17 (mod 32);P3 =8>><>>:< 2;54¡ 14m1=4 +14m1=2 ¡ 14m3=4 >; if m ´ 1 (mod 32);< 2;34¡ 54m1=4 +34m1=2 ¡ 14m3=4 >; if m ´ 17 (mod 32):2. Proof of TheoremLet L = Q(m1=2) so that Q ½ L ½ K and [L : Q] = 2. SetQ1 =< 2;1 +m1=22>; Q2 =< 2;1¡m1=22> :Q1 and Q2 are distinct prime ideals of OL such that < 2 >= Q1Q2. Letm2 be the largest integer such that m22 j m. Set m1 = m=m22 so that m1is a squarefree integer having the same sign as m. Clearly m1=2 = m2m1=21 .ThenQ1 =8>><>>:< 2;1 +m1=212>; if m2 ´ 1 (mod 4);< 2;1¡m1=212>; if m2 ´ 3 (mod 4):Next, by [2, Table D, cases D1, D2, p. 92], we see thatQ1 = P 21for some prime ideal P1 of OK . We claim thatP1 =< 2;32+m1=4 +12m1=2 > :First we show that P1 is a prime ideal of OK . The minimal polynomial ofµ =32+m1=4 +12m1=2 over Q isg(x) = x4 ¡ 6x3 + (13¡ 8k)x2 + (¡14¡ 8k)x+ (6 + 16k + 16k2):Hence N(µ) = §(6 + 16k + 16k2) ´ 2 (mod 4). Let < µ >= S1S2 ¢ ¢ ¢Srbe the prime ideal factorization of < µ > in OK . Hence N(< µ >) =2Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 5http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/5FACTORIZATION OF 2 45N(S1)N(S2) ¢ ¢ ¢N(Sr). As 2 k N(< µ >) there exists a unique S = Si suchthat 2 k N(S), that is N(S) = 2. Thus < µ > has exactly one prime idealto exponent 1 in its prime factorization lying above 2. As P1 =< 2; µ > wededuce that P1 = S so that P1 is a prime ideal of OK . Next we show thatP1 j Q1. We set Á = 32 ¡m1=4 +12m1=2. An easy calculation shows that1 +m1=22= µÁ¡ (2k + 1)2:Hence, as 2 2 P1 and µ 2 P1, we deduce that 1 +m1=222 P1. Thus we haveQ1 =< 2;1 +m1=22>µ P1, and so P1 j Q1. As Q1 is the square of a primeideal in OK , we deduce that Q1 = P 21 as asserted.Letk =½2g; if m ´ 1 (mod 32);2g + 1; if m ´ 17 (mod 32):For ² = §1, the minimal polynomial of®(²) =8>><>>:54+²4m1=4 +14m1=2 +²4m3=4; if m ´ 1 (mod 32);34+5²4m1=4 +34m1=2 +²4m3=4; if m ´ 17 (mod 32);isx4 ¡ 5x3 + (9¡ 12g)x2 + (¡7 + 24g ¡ 64g2)x+ (2¡ 12g + 64g2 ¡ 128g3);if m ´ 1 (mod 32), andx4 ¡ 3x3 + (¡37¡ 76g)x2 + (¡75¡ 240g ¡ 192g2)x+(¡38¡ 172g ¡ 256g2 ¡ 128g3);if m ´ 17 (mod 32). Clearly N(®(²)) ´ 2 (mod 4) in both cases, andsimilarly to the argument above, we deduce that I+ =< 2; ®(1) > andI¡ =< 2; ®(¡1) > are conjugate prime ideals of OK lying above 2. If m ´ 1(mod 32) we have1¡m1=22= 2(1¡ g ¡ gm1=2)¡ ®(1)®(¡1) 2 I+ \ I¡and if m ´ 17 (mod 32)1¡m1=22= 2(¡g ¡ (1 + g)m1=2)¡ ®(1)®(¡1) 2 I+ \ I¡:3Spearman and Williams: The Prime Ideal Factorization of 2 in Pure Quartic Fields withProduced by The Berkeley Electronic Press, 200646 B. K. SPEARMAN AND K. S. WILLIAMSHence1¡m1=222 I+ \ I¡. Thus I+ and I¡ are conjugate prime ideals ofOK lying above the prime ideal Q2 of OL. As < 2 >= P 21P2P3 = Q1Q2 andQ1 = P 21 , we see that Q2 = P2P3 and that we can takeP2 = I+ =< 2; ®(1) >andP3 = I¡ =< 2; ®(¡1) > :This completes the proof.References[1] T. FUNAKURA, On integral bases of pure quartic ¯elds, Math. J. Okayama Univ. 26(1984), 27-41.[2] J. G. HUARD, B. K. SPEARMAN and K. S. WILLIAMS, Integral bases for quartic¯elds with quadratic sub¯elds, J. Number Theory 51 (1995), 87-102.Blair K. SpearmanDepartment of Mathematics and StatisticsUniversity of British Columbia OkanaganKelowna, B.C. Canada V1V 1V7e-mail address: blair.spearman@ubc.caKenneth S. WilliamsSchool of Mathematics and StatisticsCarleton UniversityOttawa, Ontario, Canada K1S 5B6e-mail address: kwilliam@connect.carleton.ca(Received May 20, 2005 )4Mathematical Journal of Okayama University, Vol. 48 [2006], Iss. 1, Art. 5http://escholarship.lib.okayama-u.ac.jp/mjou/vol48/iss1/5

The Prime Ideal Factorization of 2 in Pure Quartic Fields with Index 2

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