a theorem on the norm group of a finite extension field

23. A theorem on the norm group of a finite extension field.

By Tadasi NAKAYAMA

(Received February 19, 1943.)

As an application of the relationship between factor sets and norm

groups studied formerly by Y. Akizuki(1) and the writer,(2) we prove

in the present paper a theorem concerning the norm group of a finite

extension field, from, which the so®called limitation theorem of the local

class field theory readily follows and which indeed clarifies, as it seems

to the writer, the algebraic content of this theorem. In fact the usual

proof to the limitation theorem(3) depends on the existence theorem and

hence fails to apply to a generalization of the local class field theory

considered recently by M. Moriya.(4.5) But our present approach remains,

like Moriya's own proof,(6) to be valid in this generalized case too. At

anyy rate, the writer hopes that the following supplements the hitherto

lack in the hypercomplex treatment of the local class field theory.(7)

•˜ 1. Factor sets and norm groups,

a limitation theorem.

When k, K, ƒ¶, etc. are certain fields, we denote by k*, K*, ƒ¶*,

etc. the multiplicaive groups of their non-zero elements. Further, the

(multiplicative) norm groups of finite extensions K/k, ƒ¶/k, etc. are

denoted by N*K/k, N*ƒ¶/k, etc. For a (finite) alois extension ƒ¶/k, with

(1) Y. Akizuki, Eine homomorphe Zuordnung der Elemente der galoisschen Gruppe zu den Elementen einer Untergruppe der Normklassengruppe, Math. Ann. 112 (1936),

p. 567.

(2) T. Nakayama, Uber die Beziehungen zwischen den Faktorensystemen und der N ormkla.ssengruppe eines galoisschen Erweiterungskorpers, Math. Ann., 112 (1936), p. 85.

(3) See for instance C. Chevalley, La theorie du symbole de restes normiques, Journ. fur Math. 169 (1933), p. 141.

(4) M. Moriya, Die Theorie der Klassenkorper im Kleinen uber diskret perfekten Korpern. I. Proc. Imp. Acad. 18 (1942), p. 39; II. ibid. p. 452; M. Moriya-T. Nakayama,

III. ibid. 19 (1943), p. 132. Cf. also 中山正,局所類體論 (1935); O.F.G. Schilling, The structure of local class field theory, Amer. Journ. Math. 60 (1938), p. 75.

(5) In thiss generalized theory, as a matter of fact, the limitation theorem is used conversely in proving a theorem which has an essential bearing for the existance theorem. See Moriya-Nakayama, l.c. Satz 4.

(6) See the second paper in (4).(7) C. Chevalley, l.c., T. Nakayama, l.c. Y. Akizuki, l.c.

878 Tadasi NAKAYAMA

its Galois group _??_', and for a factor set (a)=(aR.s) (R,S•¸_??_) of ƒ¶/k

satisfying

(1)

aR.STas.T=RS.TaTR.S. we put

(2) Fƒ¶/k(R;(a))=_??_S•¸_??_aR.S.

On letting S,T, respectively, run over _??_ in (1), and considering further, associate factor sets (b):

(3)

we saw formerly(8)

Lemma, 1. Fƒ¶/k(R;(a)) belongs, for every R•¸ _??_ to the ground

fi eld k, and

(4) R•¨Fƒ¶/k(R;(a)) mod. N*ƒ¶/k

maps _??_ homomorphi.cally(9) into the norm class group k*/N*ƒ¶/ of

ƒ¶/k; _??_ being the commutator group of _??_ Associate factor sets induce

one and the same homomorphic mapping, and in fact any change of

N*ƒ¶/k (B;(a)) by elements in N*ƒ¶/k as factors is accomplished by passim

over to a suitable associate factor set.

Let next H be a between-field of ƒ¶/k, and let _??_ be the belonging

subgroup of _??_ Denote then by (a)_??_ the factor set of ƒ¶/H consisting

of that part of (a) concerned with _??_ In letting S run over _??_ in (1)

and obtaining thus the equality IIS•¸_??_aR.ST=(IIS•¸_??_aR.s)T, we proved

further(10)

Lemma 2. For every A•¸_??_ we have

Fƒ¶/k(A;(a))=NH/k(Fƒ¶/H(A;(a)_??_)).

It is however useful to express this fact in a (rather weaker) moified

form:

Lemma 3. If factor sets (a), (a) of ƒ¶/H, ƒ¶/k are such that

((a), ƒ¶/H,_??_)•`((a), ƒ¶/k. _??_)H,

(8) Hakayama, l.c. Satz 1, 2 and 3.

(9) Isomorphically, if the exponent of the algebra ((a), ƒ¶, _??_) is equal to the degree

(ƒ¶:k) (cf. Nakayama and Akizuki, l.c.). But this finer result needs not be known in

proving our main theorem: it will be used only in a supplementary consideration in _??_ 4.

(10) Nakayama, l.c. Satz 5.

A theorem on the norm group of a finite extension field 879

then

Fƒ¶/k(A;(a))•ßNH/k(Fƒ¶/k(A;(a))) mad. N*ƒ¶/k.

for every A•¸_??_.

Proof. Since, as is well known, (a)•`(a)_??_, we have

Fƒ¶/H(A;(a))•ßFƒ¶/H(A;(a)_??_) mad. N*ƒ¶/H.

Hence, by the above lemma 2,

NH/k(Fƒ¶/H(A;(a))•ßNH/k(Fƒ¶/H(A;(a)_??_)=Fƒ¶/k(A;(a)) mod. N*ƒ¶/k.

Second proof. It is also possible to prove the lemma in a structural

way. Consider the crossed products

R=((a), ƒ¶/H,_??_)=ƒ¶+vBƒ¶+...+VCƒ¶, VBVC=VBCaB, C

and

_??_=((a), ƒ¶/k,_??_)=ƒ¶+usQ+...+uTƒ¶, USUT=ustaa,T

belonging to our factor sets (a) and (a). Since _??_H•`R, R is isomorphic

to the subalgebra of _??_ consisting of all the elements element-wise com

- mutative with H(•…ƒ¶•…_??_). Identifying R with this algebra we may

consider R itself as a subalgebra of _??_, and then

where Put for S=BP. Then

and the factor set (a')=(a's,T) belonging to (u's) is associate with (a). Here

which means Hence

for every B in _??_ Since (a')•`(a) this proves the lemma.

From these follows

Lemma 4. Let ƒ¶/k, _??_, H, _??_ and (a) be as before. If R•¸_??_, then

Fƒ¶/k(R,(a))•¸N*H/k.

Proof. Let R=R1R2; R1•¸_??_, R2•¸_??_. Then

Fƒ¶/k(R;(a))•ßFƒ¶/k(Rl;(a))Fƒ¶/k(R2;(a)) mod. N*ƒ¶/k.

880 Tadasi NAKAYAMA

Here the first factor in, the right hand side is, because of R1 E _??_', in

N*ƒ¶/k, while the second factor is in N*H/k in virtue of Lemme 2 (or

Lemma 3).

Now we obtain

Theorem 1. Let ƒ¶/k be a Galois extension. Let K, H be between-

fi elds of ƒ¶/k such that ƒ¶•†K•†H•†k, and R, _??_ be the corresponding

subgroups of _??_; 1•…R•…_??_•…_??_. Assume that every algebra-class over H

having _??_ as a splitting geld is obtained from a suitable algebra-class over

k by the extension H/k of coefficient field, and moreover every non-zero

element y in H can be expressed in a form

(5) ƒÁ•ß_??_iFƒ¶/H(Ai;(a)i) mod. N*ƒ¶/H (AiE_??_)

with suitable factor sets (a)i of ƒ¶/H and Ai in _??_. If then H contains

the maximal ahelian subfiel of K/k, we have

N*K/K=N*H/K,

namely, every element in k which is a norm q f H/k is also a norm of K/k.

Proof. Let c•¸N*H/k and c=NH/k(ƒÁ). By our assumption ƒÁ may be

expressed in the form (5),(11) and there are factor sets (a)i of ƒ¶/k

such that

Then by Lemma 3

mod

But Ai •¸_??_, and that H contains the maximal abelian subield of K/k

means _??_•…_??_R'. Thus c•¸ N*H/k because of Lemma 4.

Corollary.(12) Let ƒ¶/k be a Galois extension and H be a between

field containing the maximal abelian subfield of ƒ¶/k. Let there exist a

series of fields ƒ¶=H0>K1> ... >Km=H such that Ki/k are all Galois

extensions and Ki/Ki+1 are all cyclic (i =0, 1, m-1). Assume that

every algebra-class over Ki having ƒ¶ as a splitting field is obtained from

an algebra-class over k. Them N*ƒ¶/k=N*H/k.

Proof. This is trivial in case m=0. Suppose therefore m•†1. For

every ƒÁ in K*1 we have ƒÁ=Fƒ¶/k1(S;(ƒÁ)), where S is a generating auto

- morphism of ƒ¶/K1 and (ƒÁ) denotes the factor set of the cyclic algebra

(ƒÁ, ƒ¶/K1, S). Thus N*ƒ¶/k=N*K1/k by our theorem. Therefore, if we know

(11) Cf. the final remark in Lemma 1.

(12) The corollary applies for instance to a p-adic number field k, its Galois extension ƒ¶ without higher ramification, and the maximal abelian subfield H of ƒ¶/k.

A theorem on the norm groap of a flnite extension field 881

already N*K1/k=N*H/k, then we would have N*ƒ¶/k=N*H/k. Since K1 satisfies

the same conditions as ƒ¶, this proves our corollary by induction with

respect to m.

•˜ 2. On solvable extensions.

Although Theorem 1 is already of use, (13) its assumption concern

-ing the form (5) of ƒÁ is rather awkward. We want therefore to proceed

further to replace it by a more natural one. To do so we need Akizuki's(14)

Lemma 5. Let H be a alois between-field of a Galois extension

ƒ¶/k, _??_ be the corresponding invariant subgroup of the Galois group

_??_ of ƒ¶/k. Let (ƒ¶:H)=h. For a factor set (a) of ƒ¶/k, put

(6)

where {P0, Q0,...} denotes a system of representatives of _??_/_??_ and P denotes the class (co-set) of P med. _??_. Then (a)=(ap,Q) is a factor set of H/k and

(7)

Further

(8)

Leaving the first half of the lemma to the papers by Witt and Akizuki's, (15) let us be contented with deriving (8) from (6). Namely, from (1) follows

whence

Therefore

which proves (8).

Lemma 6. Let ƒ¶/k, _??_, H, _??_ be as above. Suppose that every ƒÁ

in H* may be expressed in the form (5) (in •˜1), and that the algebra

- classes ((a)i,ƒ¶,_??_) can be obtained from algebra-classes over k by

extension of coefficient field. Let further ((b), ƒ¶, _??_)h•`((ƒÀ), H, _??_/_??_) with

(13) See •˜4.

(14) Akizuki, l.c.

(15) E. Witt, Zwei Regein uber verschrankte Produktc, Journ. fur Math. 173 (1935),

p. 191; Y. Akizuki, l.c. •˜1.

882 Tadasi NAKAYAMA

factor sets (b), (ƒÀ) of ƒ¶/k and H/k. Then for the element

mod

here P0 being a representative of P, (a)i factor sets of ƒ¶/k, and Ai •¸_??_.

Proof. From Lemma 5

mod

Namely, is in and with

Here ƒÁ is of a form (5), and ((a)i, ƒ¶,_??_)•`((a)i, ƒ¶, _??_)H for suitable factor

sets (a)i of 9ƒ¶/k. Hence by Lemma 3

mod

Therefore

Theorem 2. Let ƒ¶/k be a solvable Galois extension, that is, its Gaioi

group be solvable. Let

where _??_i-1/_??_i are cyclic and generated by Si (•¸_??_), and let

be the corresponding field series. Assume, for each i, that every algebra

- class over k having Ki as a splitting field is an (ƒ¶:Ki)-th power of suitable

algebra-class having ƒ¶ as a splitting field, and moreover every algebra-class

over Ki possessing ƒ¶ as a splitting field is obtained from a one over k by

the extension Ki/k of ground field. Then, for every non-zero element c in

k there exist n factor sets (a)i of ƒ¶/k such that

mod

Proof. The case n=0 is trivial. So is the case n=1 too, for

where (c) denotes the factor set of the cyclic crossed product (c, K1, S1).

To prove our theorem by induction, assume that it is true for n-1

instead of n. Then, since ƒ¶/K1 satisfies the same conditions as 9ƒ¶/k,

there are, for every ƒÁ1 •¸ K1, n-1 factor sets (a)i such that

mod.


Here, by our assumption, ((a)i, ƒ¶,_??_1)•`((a)i, ƒ¶,_??_)k1 with suitable

factor sets (a)i (i=2, 3, ...,n) of ƒ¶/k,

Now., let c•¸k*. c=FK1/k(S1;(c)), as mentioned above. By our

assumption there is a factor set (a)1 of ƒ¶/k such that the class (cl, K1, S1)

is the (ƒ¶:K1)-th power of the class ((a)1, ƒ¶,_??_). Thus, from the above

observation and Lemma 5, c is expressed in the form

mod.

This proves the theorem.

•˜ 3. Main theorem.

Now we come to our maim result;

Theorem 3. Let K/k be a finite extension, and . L be its subfield

containing its maximal abelian subield. Let ƒ¶/k be the Galois field of

K/k, and assuwe that ƒ¶/L is solvable, Moreover, suppose for every field

A between ƒ¶ and L that every algebra-class over L having A as a splitting

field is an (ƒ¶:A)-th power of an algebra-class hawing ƒ¶ as a splitting

fi eld, and that every algebra-class over A possessing ƒ¶ as a splitting field

is obtained from an algebra-class over k by extension of ground field. Then

Proof. According to Theorem 2, every ƒÁ in L can be expressed in

a form ƒÁ•ßIIiFƒ¶/k(Ai;(a)i) mod. N*ƒ¶/L. Then, by Theorem 1, N*k/k=N*L//k.

•˜ 4. On the limitation theorem of the local

class field theory.

It is evident that the limitation theorem of the local class field

theory, including the generalized case,(16) follows immediately from our

Theorem 3, for the assumptions of the theorem are satisfied there.

However, for the mere purpose of proving the limitation theorem,

already Theorem 1 is sufficient, when combined with a previous result in

Nakayamaa, l.e. and Akizuki, l.c. (17):if the crossed product ((a), ƒ¶/k, _??_)

has the exact exponent (ƒ¶: k), then the correspondence (4) in Lemma 1

gives an isomorphism of _??_/_??_ with a subgroup of the norm-class group

k*/N*ƒ¶/k. Namely, let k be a p-adic number field (or a complete valuated

(16) See footnote (4).

(17) Akizuki, l.c., •˜2. For the proof, combine, namely, Satz 6 in Nakayama. l.c. with Lemma 5 above.

884 Tadasi NAKAYAMA

field, whose residue class field is a perfect field possessing for each

natural number m a unique extension of m-th degree), and consider at

first a Galois extension ƒ¶ over k. Denoting the maximal abelian subfield

of ƒ¶/k by A, we show that N*ƒ¶/k=N*A/k. To do so, assume that the

assertion is the case for extensions of smaller degrees. Then N*ƒ¶/A=N*A1/A,

when A1 denotes the maximal abelian subfield of ƒ¶/A. Hence

being the Galois group of ƒ¶/k),

and this implies, in view of the above mentioned result, that every

element ƒÁ in A can be expressed in the form ƒÁ•ßFƒ¶/A(A;(a)) mod. N*ƒ¶/A

(A•¸_??_'), when ((a), ƒ¶/A, _??_') is a division algebra. Therefore, Theorem

1 can be applied to show that N*ƒ¶/k=N*A/k. Our assertion is proved

thus by induction. Now, consider a finite extension .K/k over k, and a

between-field L containing its maximal abelian subfield. Let ƒ¶/k be

the Galois field of K/k. Apply then the above observation to ƒ¶/L, instead

of ƒ¶/k, which shows, again combined with the result mentioned above,

that every element ƒÁ in L has a form ƒÁ•ßFƒ¶/L(A; (a)) mod. N*ƒ¶/L. Hence

we have N*K/k=N*L/k, again because of Theorem 1, and this proves the

limitation theorem.

•˜ 5. An, example.

Let us finally observe an example of a case, to which our theorem 3,

algebraic "limitation theorem", may be applied, though rather trivial,

and in which norm-group-indices behave quite differently from the case

of local class field theory.

Let P. Q be the fields of real and complex numbers, respectively,

and P(x), Q(x) the rational function fields over them. Let further be the field obtained from Q(x) by adjoining the third roots of x. Then

is a Galois extension of degree 5 with the maximal abelian

subfield Q(x). Because there is no non-commutative division algebra

ever Q(x),(18) our theorem 3 may be applied to k=P(x),So

Of course this is trivial since N~(s ),p()=Q(x)*, We are only interested in that here the norm®rou.p-indices are quite different from the case of p-adic number fields. Namely

(18) This is a (rather trivial) special case of a theorem of Tsen.


For, an element (•‚0) in P(x) has a form

with mutually distinct linear polynomials f1, f2, ... , f3 and irreducible

quadratic polynomials g1, g2, ..., gt. And, ƒÁ(x) is a norm of Q(x)/P(x),

if and only if u's are all even, as one readily sees.

Department of Mathematics,

Nagoya Imperial University.

a theorem on the norm group of a finite extension field

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