is divisible by ~ i f l ~ j.

By

LADISLAS FUCHS in BUDAPEST.

I n his p a p e r " S ~ t z e iiber a l g e b r a i s c h e R i n g e ''1 T. N a g e l l has discussed c e r t a i n p r o p e r t i e s of algebraic rings. T h e p r e s e n t n o t e c o n c e r n s itself with t h e g e n e r a l i z a t i o n of these results to r e l a t i v e a l g e b r a i c r i n g s ; t h e t h e o r e m s will be t r a n s f e r r e d w i t h o u t e s s e n t i a l c h a n g e .

I n w h a t follows we shall m e a n by F a finite algebraic n u m b e r field a n d by R t h e r i n g of t h e i n t e g r a l e l e m e n t s of F . L e t f u r t h e r ~ be an algebraic field o v e r F of d e g r e e n a n d l e t P be t h e r i n g of t h e i n t e g r a l e l e m e n t s of ~. I t is well k n o w n t h a t in ~ t h e r e are n e l e m e n t s z, col, 9 9 con, b e i n g l i n e a r l y i n d e p e n d e n t w i t h r e s p e c t to F , such t h a t e v e r y e l e m e n t of ~ possesses a u n i q u e r e p r e s e n t a - t i o n of t h e f o r m

co = alcol + " " + an con (I)

w i t h coefficients in F . T h e coi are called t h e basis of ~ w i t h r e s p e c t to F . L e t be a n e l e m e n t of P of t h e e x a c t d e g r e e n, t h a t is, ~ is a r o o t of an i r r e d u c i b l e algebraic e q u a t i o n x '~ + r I x ~-1 + . . . + r n - - - o w h e r e r~. are in R. I n view of (I) we m a y set

= e lcol + . . . + ( 2 )

f o r k = o , I . . . . , n - - I . Since ~ was c h o s e n so as to be of t h e e x a c t d e g r e e n, t h e d e t e r m i n a n t c----Icktl of t h e coefficients in (2) does n o t vanish, a n d so t h e system m a y be i n v e r t e d , a n d t h e n we g e t

co, = - (b,1 + b,~ ~ + . . . + b,,, ~"-1), 1 (b,~, F ) (3) c.

f o r i---- I, 2, . . . , n.

19

i Math. Zeitsehrift 34 (1932), PP. 179--I82.

2 The elements of F will be denoted by Latin, those of ~ by Greek letters.

F o r the sake of convenience we suppose t h a t t h e r were so chosen t h a t whenever ~ in (I) is integer, t h e a~ are all integers, i. e., are all i n / / . . T h e n so are of course t h e e~i in (2) [and hence c] as well as the b~ in (~).

On a c c o u n t of (I) a n d (~) one sees at once t h a t

~o = - a,(b,x + b , , ~ + -., + b , . ~ "-~) = I { (~a,b,a) + ... ~- _ (:~a~ b,,,) ~"-x},

l = 1

t h a t is to say, by m e a n s of the powers of ~ every element of P has a representa- tion of the f o r m

= -(c~ + e, ~ + - . + c~ ~ - ~ ) , I (e,~ R). (4)

e

(4) is unique in ct, f o r I, ~, . . . , ~ - 1 are linearly i n d e p e n d e n t with respect to R.

L e t n o w P* be a subring of P c o n t a i n i n g ~. Every e l e m e n t ~ of P* m a y clearly be represented in the f o r m

7 = ~ ( c 1 + e 2 _ ~ + ' " I + c ~ - 1 ) , (c~,R, 1 < l ~ n )

where cz ~= o. Consider all t h e 7 for a fixed n u m b e r 1. I t is easily seen t h a t t h e last coefficients s cz c o n s t i t u t e an ideal in //. T h a t t h i s ideal ~z m u s t con- t a i n a non-vanishing e l e m e n t a n d so ~z is d i s t i n c t from t h e zero-ideal, is evident.

S e t t i n g ~z=(ct(j) . . . . , czc~z)}, it is also e v i d e n t t h a t to each basis element c~(~;

there corresponds a n u m b e r 7~ ~) of P" w i t h t h e last coefficient e~):

y~l = ~_ ~el~ + c ~ ~ + . , . + c ~ ~ - ' ) _{c X l l}

(5) c!,l ~ R, 13 ~ 1 = e~l, ~ll ~ ~ I , . . .~ m l ) .

The elements 7~ II, 9 " ., 7~,}, ~,cll t2 , " ", 7~ ~'), . . . . ., }'~), ., 7~m~ ), or, if we w a n t to have t h e indices r u n n i n g successively f r o m I u n t i l h r ~ ~ ~z, t h e elements 7 z , . . . , ~,v

l--1

f o r m a basis of P* with respect t o R, t h a t is to say, eve~7/ element o f P* can be expressed i n the f o r m

7 --- d, Y, + " " + dzr 7~', (d, ~ R). (6) However, this r e p r e s e n t a t i o n is n o t unique, in general.

s M o r e p r e c i s e l y : t h e c - t i m e s o f t h e l a s t c o e f f i c i e n t s .

The powers of ~ are in P*, we can therefore find numbers x of R such t h a t for k > I

k

~ - 1 = y, (~i,)~ill + ... +

xl,.,,r~.~)), (~J ~). (7)

I f we replace here ft a) by their values taken from (i), one sees immediately t h a t the coefficient of ~ - 1 is I on the left side, while on the right side

v ~ k k C

ck being a number of ~ . From the equality of the two coefficients, implied by the linear independence o f I, ~ . . . . , ~-~, it follows ^{c ~ ok. W e} thus get that ^r is an element of every ~k(k > I):

T h e o r e m 1.

The determinant c---let,[ is divisible by ~ for k > t.

We further get from (5) the equality

~,~') 9 ~ - ~ - - -~ ~c~)~J-~ + . . - + c(~) ~ - ' ~

- - e ~ l l II /

showing that c~q) and similarly, every basis element of ~t is contained in ~ for l - - j . This implies that ~l----o(~j) for l--~j, that is in words,

T h e o r e m 2. ~t

is divisible by ~ i f l ~ j.

Let u s n o w turn our attention t o the proof of

T h e o r e m

3. cl~) is divisible by ~ .

Proof by the priric~ple of mathematical induction. F o r 1 - I the asser- tion is trivial. Let us suppose t h a t c(~ for ] c - - ~ l - - I is divisible by ~k and so a fortiori by ~_~, in accordance with theorem 2. Consider ~ ) and take an element c' of ~t-__~ The last coefficient ~ of ~ "

c'~), c'cl~)

Y~t-1 C(/~/11)

' ) - (m~-~)~

holds. Hence we conclude that c ~,~ -- ~:~( . . . . ~-~(~) + "'" § Y~-I ~t-~ ~ ~ contains only powers of ~ with exponents not greater than l - - 2 ; so t h a t we obtain

.l--1

c' ~I~)= ~ c~(~)~,) + . . . + ~ ( ~ ( ~ ) ~ + (,:, ~ ) ~ + . . . + ~(~-,) ~).

',~k r ~ k r / i t - - 1 Y m l - - 1 / I - - 1

k = l

Setting here for the ~,(~e) their values taken from (5), we see that on the right hand side the first subscripts of c~ are not greater than l - - ~ , therefore by

assumption we may hence conclude t h a t the (e-times) coefficients of the powers of ~ are divisible by ~z-~. The fact t h a t the coefficients of the same powers of ~ must be equal on the two sides implies t h a t c'e~ ) - - o(~1-1). Since c' was arbitrary in --~-, we finally get t h a t ~j~(~) must be contained in ~t, and this completely establishes the theorem.

We now pass to the proof of the following theorem.

T h e o r e m 4. The

relative diseriminant of P* with respect to R:

a.,/B

where 1)(~) is the relative d i s ~ m i . a n t of ~.

All the determinants of order n of the matrix 4

(8)

....... o . . .

generate an ideal ~* in a Galois-overfield of F containing 4. The square of ~*

is an ideal in R and is equal to the relative discriminant of P* with respect to R. ~* may easily be verified to be the ~-times product of I

I I . . . I

9 ,~ . .. . . . . . . . . . . .

~ ( 1 ) , - 1 ~ ( ~ ) , - 1 . . . ~ ( , ) n - x

and the ideal ~ generated by the n.ordered determinants of cn e~l . . . . e~-i ]

I

et~ e2z . . . . e , v 2

~ l n e 2 n . . . . C N n

9 7(i) is t h e i t h c o n j u g a t e of ~ .

where t h e c,~ are t h e coefficients for which

- C~,i ~ i - 1

(cf. ~5); some of c,~ are vanishing). As I have proved elsewhere 5, ~ is equal to t h e idealproduct ~ j . . . ~,~, so t h a t we are led to t h e r e s u l t e n u n c i a t e d in t h e o r e m 4.

6 A t h e o r e m on t h e r e l a t i v e n o r m of a n ideal. C o m m e n t a r i i M a t h . H e l v e t i e i 21 (i948}, pp.

2 9 - - 4 3 ; see t h e o r e m I.

Y

37--48173. Aeta mathematiea. 81. Imprim6 le 29 avril 1949.