MATHEMATICS

THE CARTAN-BRAUER-HUA THEOREM

BY

JAN TREUR

(Communicated by Prof. H. Freudenthal at the meeting of January 29, 1977)

In this note we present an elementary proof which hardly uses any

computation; cf. [a] p. 2. First a lemma.

LEMMA:

Suppose K is a skew field and E is a left K-linear space.

Assume M C E is closed under addition and contains at least two inde-

pendent elements over K. Let rj: M -+ E be

a

map satifying $(a+ b) =

= d(a) +4(b) and &a) = 31cc a for all a, b E M and certain ila E K. Then

there exists one element 1 E K such that $~(a)=Ja for all a E M.

PROOF:

Let a,

b

E M be nonzero ; choose c E M such that a and c are

independent. From &a + A& =+(a) + 4(c) = +(a + c) = jla+Ja + c) it follows

that A,=1 a+c=Jc. Since

b

is independent of a or c, also jib = 3La = 1,.

THEOREM

(Cartan-Brauer-Hua) : Suppose L/K is an extension of skew

fields and M C L is closed under addition and 1 E M. If aK C Ka for all

a E M then either M C K or M C ZL(K) (h ere ZL(K) denotes the centralizer

of K in L). In particular, if aKa-l= K for all nonzero u E L then either

K = L or K is contained in the center of L.

PROOF:

Suppose M is not contained in K, then M contains at least

two elements left independent over K. Let

b E

K be given; for each

a

E M,

ab E aK C Ka. Hence ab=&a for some 1, E K. Apply the lemma to the

right multiplication rb : a I+- ab : M -+ L. We have an element il E K such

that ab=la for all a E M. In particular this holds for a=

1,

hence 2= b.

Therefore ab= ba for all a E M and b E K. For the last statement, take

M=L.

REMARK:

The above proof remains valid if L is replaced by any ring

extension of the skew field K.

Mathematkch Inetituut

De Uithof, Utrecht

454

REFERENCES

1. Cartan, H. - Theorie de Galois pour les corps non commutatifs, Ann. Sci. Ecole

Norm. Sup. 64, 59-77 (1947).

2. Brauer, R. - On a theorem of H. Cartan, Bull. Amer. Math. Sot. 55, 619-620

(1949).

3. Hua, L. - Some properties of a sfield, Proc. Nat. Acad. Sci. USA 35, 533-537

(1949).

4. Treur, J. - A duality for skew field extensions, Dissertation, Utrecht, 1976.