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, Ħ, etc. are certain fields, we denote by k*, K*, Ħ*,
etc. the multiplicaive groups of their non-zero elements. Further, the
(multiplicative) norm groups of finite extensions K/k, Ħ/k, etc. are
denoted by N*K/k, N*Ħ/k, etc. For a (finite) alois extension Ħ/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*Ħ/ of
Ħ/k; _??_ being the commutator group of _??_ Associate factor sets induce
one and the same homomorphic mapping, and in fact any change of
N*Ħ/k (B;(a)) by elements in N*Ħ/k as factors is accomplished by passim
over to a suitable associate factor set.
Let next H be a between-field of Ħ/k, and let _??_ be the belonging
subgroup of _??_ Denote then by (a)_??_ the factor set of Ħ/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
FĦ/k(A;(a))=NH/k(FĦ/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 Ħ/H, Ħ/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), Ħ, _??_) is equal to the degree
(Ħ: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), Ħ/H,_??_)=Ħ+vBĦ+...+VCĦ, VBVC=VBCaB, C
and
_??_=((a), Ħ/k,_??_)=Ħ+usQ+...+uTĦ, 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*Ħ/k, while the second factor is in N*H/k in virtue of Lemme 2 (or
Lemma 3).
Now we obtain
Theorem 1. Let Ħ/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 Ħ/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 Ħ/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 Ħ/k be a Galois extension and H be a between
field containing the maximal abelian subfield of Ħ/k. Let there exist a
series of fields Ħ=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 Ħ as a splitting field is obtained from
an algebra-class over k. Them N*Ħ/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 ex- tension Ħ without higher ramification, and the maximal abelian subfield H of Ħ/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*Ħ/k=N*H/k. Since K1 satisfies
the same conditions as Ħ, 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
Ħ/k, _??_ be the corresponding invariant subgroup of the Galois group
_??_ of Ħ/k. Let (Ħ:H)=h. For a factor set (a) of Ħ/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,Ħ,_??_) 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 9Ħ/k. Hence by Lemma 3
mod
Therefore
Theorem 2. Let Ħ/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 (Ħ:Ki)-th power of suitable
algebra-class having Ħ as a splitting field, and moreover every algebra-class
over Ki possessing Ħ 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 Ħ/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 Ħ/K1 satisfies the same conditions as 9Ħ/k,
there are, for every ƒÁ1 •¸ K1, n-1 factor sets (a)i such that
mod.
A theorem on the norm group of a finite extension field 883
Here, by our assumption, ((a)i, ƒ¶,_??_1)•`((a)i, ƒ¶,_??_)k1 with suitable
factor sets (a)i (i=2, 3, ...,n) of Ħ/k,
Now., let c•¸k*. c=FK1/k(S1;(c)), as mentioned above. By our
assumption there is a factor set (a)1 of Ħ/k such that the class (cl, K1, S1)
is the (Ħ:K1)-th power of the class ((a)1, Ħ,_??_). 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 Ħ/k be the Galois field of
K/k, and assuwe that Ħ/L is solvable, Moreover, suppose for every field
A between Ħ and L that every algebra-class over L having A as a splitting
field is an (Ħ:A)-th power of an algebra-class hawing Ħ as a splitting
fi eld, and that every algebra-class over A possessing Ħ 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), Ħ/k, _??_)
has the exact exponent (Ħ: k), then the correspondence (4) in Lemma 1
gives an isomorphism of _??_/_??_ with a subgroup of the norm-class group
k*/N*Ħ/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 Ħ over k. Denoting the maximal abelian subfield
of Ħ/k by A, we show that N*Ħ/k=N*A/k. To do so, assume that the
assertion is the case for extensions of smaller degrees. Then N*Ħ/A=N*A1/A,
when A1 denotes the maximal abelian subfield of Ħ/A. Hence
being the Galois group of Ħ/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*Ħ/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 Ħ/k be
the Galois field of K/k. Apply then the above observation to Ħ/L, instead
of Ħ/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.
A theorem on the norm group of a finite extension field 885
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.
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