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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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