Title A Remark on the Decomposition of Prime Ideals in Galois Extensions Author(s) OHTA, Kiichiro Citation [岐阜大学教養部研究報告] vol.[20] p.[7]-[11] Issue Date 1984 Rights Version 岐阜大学教養部 (Dept. of Math. Fac. of Gen. Educ., Gifu Univ.) URL http://repository.lib.gifu-u.ac.jp/handle/123456789/47552 ※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。 7 A R emark on the D (?composition of Prime ldeals in Galois E χtensions Dedicated to Professor K . Y amamoto on his 60th birthday K iichiro OH T A D ept. of M ath. , F ac. of G en. E duc. , Gifu U niv. (Received Oct. 1. 12, 1984) l ntroduction. L et ん be an algebraic number field of finite degree and we denote by o the ring of integers of ヵ. 二L et F(X ) be a m onic irreducible polynomial of degree 77 1n o[χ], whose discriminant we denote by Do Let 尺 bethesplitting field oI F(X) over ん and we put 朋 = [尺 : ん ]. T hroughout in this paper we assume that p is a prime ideal of ん which is prime to 2D . T hen, it is easily seen in our case that p is un- ramified in 尺/ 力 and hence the decomposition of p in 尺 is of type (1) p= 畢邱 2●●・畢g, NKlk障 j) = 丿 ( y= 1, 2, ‥. , g), g/ = 聊・ - N eχt, we denote by F(X ) the image of 7吊り by the canonical homomorphism o→ - o/p and by り the degree of the greatest common divisor of F6り and χ刮心一χin o/p[χ] for f = 1, 2 respectively. Here, it is easily seen that in the factorization - of 耳 幻 as a produd of irreducible polynomials in o/p[χ] the number of linear factors is equal to り and also the num ber of quadratic factors is equal to ( り ー り ) / 2. Finally according as the residue clasS D mod p is a square or not in o/ p we put χμ刀 ニ フ or - 7 respectively, N ow, in this paper we shall show that if we have 万一 r1 ≦ 7; then in the decomposition of p in 瓦 as ( 1) the inertial degree y and hence the decomposition number g = mZf are uniquely determined by at most り , り and χ限り. Finally, w e shall deal w ith the case where 尺 is an 瓦 -eχtensi(jn of the rational num ber field Q for 72 = 5, び and 7; and using our results we shall consider a law of decomposition of (♪) in 尺 , where /・ is a rational prime number prime to 2D . 2. P reliminaries. B efore we state our m ain theorem , w e need the following tw o lemmas, first of which we are due to R , G . Sw an [31. L EM MA 1. (Sw an) L d m) tatiolls alld ass14mptioMs be as abow . μ s is lhe llMm bey 一 of 加 e血 c濯 e血doys of F(X) il八 Zや[X ], theu uJe haむe sE M (mod 2) ぜ 皿 d O戒y 汀 れZ刀= 7. 8 K iichiro OH T A L EM MA 2. ・L d MoM iolls 皿 d assum j) tioMs be as 岫 o叱 . - 一 一 lf - (2j F (X) = GI(X)G2(X) ‥ . GS(X) is tk 面 d E ible 血 do疸 d o of F (X ) in 9ノや[X 秘 nd j ,・ is tk degyee of G i(X ) か ら 一 2 , ‥ ・ , s yesj)ec白 雨 バ ha m oM m ㎡ tφ l e of f l , 几 , 一 = 7, the in出 id 白 砂 e f of や 緬 K yk is 岡 回 八 o the k st com 一 ‥ . J S. PROOF. Let 牢 be one of the prime factors of pin 尺 and 瓦 be the decomposition field of 叩 in 尺/ た M oreover, let ら be the 雫-adic number field. be the p-adic number field and also 尺l Since 叩 is unramified in our case, both K ZK o and 尺丿聡 arecycliceχtensionsof degreeズ Next, it is easily seen from HensePs lemma that corresponding to (2) we have the irredud ble factorization of 程 幻 in 胎 [χ ] as fo110w ing ; 芦蜀 = GI(X)G2(X) . ‥ GS(X), where the degree of G ぶり is equal to 八 for f= I , 2 , ‥ . , s resped ively. M ore- over, if ∂J s a root of G て幻 = θ for f= 1, 2 , . ‥ , s, then the field 臨 陥 ・y is a cyclic extension of degree 石 of 垢 obtained by root of unity to 垢 . adjoining N ow , as w e have K P = 臨 ( Ob the primitive ( yW p戸 - 1)th 021 ‥ 。 OS) clearly, it follow s immediately that/ = [KP : 垢] is equal to the least common multiple of ム ‥ ・ , ん 3. M ain theorem. 0 ur main theorem is as following ; T HEOREM 1. L d Mold i皿 s aRd αssum邱 皿 s be as αbow . thm the 泌e妬d d昭紀ej of や泌 K yk is Muiqw l:y ddeym緬d lf lノ ue hα叱 三 社- り ≦ 乙 by at most yl, γ 2 α祖 χ(D) as翔o面昭 励 le shouX s: - 7 7 - 7 7 2 jに 2 4 5 6 3 - PROOF. づ 7 j ソT づ θ 丿 一7 14 7 仇 - yl) ノ2 δ び G 6 N 5 θ 5 J 2 4 θ び χμ刀 4 7 jに Z 2 Q P gS CI θ 刀 一 yj θ 72 Z づ 7 j θ - L et H (X ) be the greatest com m on divisor of 耳 Uり 一 一 一 and X ″(o - X in 一 〇 /p[X ] and we put F(X) = H(X) T(X). Then, H(X) is a produd of り factors of de- gree7 1n (2) and 7¯ `μンhasnolinear factors in o /吋X] clearly. Now, it is easily seen from Lemma 2 that to prove our theorem it is suffident to consider the irre- ducible faは orization of T の in o/ p[X I . A s it is trivial for each case where μ一 り ≦ J, (a) hence we may assume 刄一 り ≧ 4. The case 刀一り = 4。 As w e have r 7≡ μ (mod2 ) in our case clearly, using L emma l it is easily verified that if χの 片 7 then T (X 八 s a product of tw o irreducible quadratic polynomials, and - if otherW ise then 7μ ワ is an irreducible polynomial of degree 〕4 1n o/ p[X ] . N ow , from L em m a 2 0ur assertion follows im m ediately for our case. (b) The case iz- り = 5。 - As り 幸77 (mod 2) in our case, if χμ刀= 7 then T(X) is an irreducible polyno- A Remark on the Decomposition of Prime ldeals in Galois E xtensions 9 - mial of degree 5 ・, and if otherwise then 7y幻 is a pr6duct of two irreducible polynomials of degree 2 and j respectively. N ow , our assertion follows immediately from L emma 2. (c) The case μ一り = 6. A S γl ≡ 以 m od 2 ) in our case, if χ μ刀 = 7 then T(X パ s decomposed into a pro。 - - - dud of tw o irreducible polynomials in o/ p[χ ] such that 耳 刃 = U(X ) V(X ) , where we - - - may assum deg び (X ) ≧deg F (X ). M oreover if ( 心 一 心 ) / 2 = θthen we have degび (X ) _ _ = degF (X ) = J, and if ( 心 - n ) / 2 = 7 then we have degF (X ) = べ y dearly。 - Next, if χμ刀= - 7, then according as ( り ーり )2 = θor 3 T(X) is an irredudble polynomial of degree び or a produd of three irreducible quadratic polynomjals n o/ p[X j. N ow , our assertion also holds in this case. (d) The case 77- り = 乙 As り 幸双 (mod 2 ) in our case, if χμ刀= 7, then according as (り 一り)/ 2 = θ or - 2 7μ ンis an irreducible polynomial of degree Z or a produd of three irreducible polynom ials such that tw o factors are quadratic and the remains is cubic. 0 n the - other hand, if χ俵り こ - 7 then 7y幻 is decomposed into a produd of tw o irredud ble - - - - polynomials in o/ p[X ] such that T (X ) こ U (X ) V ( X ) , where w e may assum e degU ( X ) - ≦degF 俵し ズ Then it is easily seen that according as ( り ーり) ノ2 こ 0 0r 7 we have 一 - degU (X ) = 2 and degF 区 り= j respectively. Now, our assertion also follows from Lemma 2 1n thjs case. T hus, our theorem is proved completely. N ow, as the immediate cnsequence of T heorem l we have the following ; T HEOREM 2. L d m ぱ i皿 s aRd 心 sltm 師 皿 s be as 晶 o叱 , 仔 11≦ 7, 匝 四 知 y ew 砂 函 meideaり of k 函 md o2D 回 can d由 ymine tk 励 eof decompo涵 ou of い n K as(l) 胎 m池鏑g Mse of d 琲ost yz, y2 alld χC D) , Pay面辺のすy, ぴ 71≦5 /ノ認肴wecαルdo 1124si昭 O戒y yl 皿 d χ(D) . 4. T he decomposition of ( /・) in som e 瓦 -eχtensions of Q. N oW, in this section we shall deal w ith the special case where カ is the rational number field Q and hence a is the ring of rational integers Z . lt is clear in our case that if 夕is a rational prime number prime to 2Z:) , then in above we may take the R egendre s symbol ( 一 多 ) in place of χ の J. M oreover xve assunle 尺 is an 刄 -extension of Q , that is, 瓦 is a Galois num ber field over Q such that the Galois group GT尺/ 剛 js isomorphic to the symmetric group Sn of degree 歓 As it is trivial for 刀≦ べ / , in follow ing w e m ay restrict our consideration to the cases where 77= 5 , 6 and Z. T hen, from above theorenls xve can sunlnlarize our consideration as follow, ● mg・ し Let 耳ミ幻 be a monic irreducibl(j polynomial of degree 771n Z [χ], where we assume tz= 5, 6 and ZI Let 刀 be the discriminant of 耳 λフ and A7 be the splitting field oI F (X ) over Q , where we assume 瓦 is an 瓦 -eχtension of Q . M oreover let - /) be a rational prime number prime to 2D and F (X ) be the reduced polynomial of K iichiro OH T A 10 - - F(X) by mod ♪. ・ lf we put り = deg (F(X) , X I) - X ) and り = deg (F(X) , XI)2- X ) in ZZ(p) [割 respectively, then in the decomposition of ㈲ 垣 尺 such that (勿 二pjp2 ‥ . pg, 皿 ら)ニダ (i= l , 2 , ‥ ・, g), g/z= μ!, both the inertial degree y and the decomposition number g = が が of ㈲ are uniquely determ ined by at m ost 巧, 乃 and G 1) as following ; (1) The case tz= 5. (i) lf り = 5, then / = 7 and g= 120. 一 1 ぐ11 j li eitheTylこ 3 01` り⊃j 311d (夕・) = 7 , then / = 2 and g = びθ (iii) ll yl二2 , then / べ ? and g= 祁 . ( iv) lf り = 7 and ( タ ・) = - j , then / = 4 and g= 30、 (v) lfり゛θ 31 1d(ク・)=1, th e 1 1ノニ52 1 1 1dg二 政 /. 臨) lfり二 θ 叩d(夕 ・)二 ¬7,th e n/=ぢa n dg=20. (2) The case jl= 6レ O ) ll yl〒6 , then / = 7 and g= Z2θ. ( ii ) lf we have one of the following ; (a ) り=4, (b ) り=?a n d(ダ)= 1八C) り = 0 , y2二 6 and ( y ) 二T j then ダ= 2 and g= 36θ. (iii) lf either り = J or り = り = θand ( ダ ) = j , then / = X and g= 240. (功 lf either り 二2 ゛ ld ( グ ) = - 1 0T yl= 0 , y2= 2 and ( ク・) = 1, thell / ゜ 4 311d g = 7 8 θ. (v)¥ lf万戸j 3111! (ダ) 二l , the11ノニ5 3ndg二144. (vi) lf eitりer り = 7 叩 b ( ダ ) = - 7 0r り = り = θand ( ダ ) = - 7, then y= び and g = 12 0 . (3卜 The case 77= Z. (i) lf り = Z, then y= 7 and g= 5040 . ( ii) lf we have one of the following ; (゛l) yl= 5 , (b) り 二J 311d ( タ ・) = 7, (C) yl= 1 , γ2= 7 an d ( 晋 ) 二 ¯ j ・ thell ノ ニ 2 and g = 2520 . (iii) lf either り = 4 0r yl- y2= 7 and ( ダ ) = 7 , then / = X and g = 1680 . ( iv) lf either り = X and ( ク・) = づ oT yl= 1 , y2= J and ( ク ・) = 7 , then / = 4 and A Remark on the Decomposition of Prime ldeals in GaIois E xtensions 11 g = 12 60 、 ( v) lf り = 2 and ( y (vi) lf we have one of the follow ing ; (a) り = 2 and ( ダ ) = j= 7, the n/= 5andg= 1008、 一 7, (b) り= り = 7 and ( ダ ) = - 7 , (c) り = 0, y2こ 4 a nd (タ・)=7, then j = 6 and g = 840 . (v ii) lfり=り=θ a n d(ダ)=j, th e n/=7a n dg=Z2θ . (viii) lf り = 0, y2= 2 and ( ダ ) = - 7, then ズ= j θand g= 50孔 (ix) ll yl= y2= θand ( ダ ) = - 7, then / = 12 and g= む θ. R eferenCeS [1] H. Kempfert, 0n the Factorization of Polynomials. J. of Number Theory, W /. 1 (1969), 116120. [2] K. 0hta, Factorization of Prime lbeals Galois Number Fields (I), (ID (in Japanese). the F ac. of Sci. and T ech. M eij o U niv. Reports of 12 ( 1971) 381-387, 招 ( 1972) 255-263. [31 R. G. Swan, Factorization of Polynomials over FiniteFields. Pacific J. of Math. 吻 /. 12 (1962) 1099-1106.

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