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# The Cartan-Brauer-Hua Theorem

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## Abstract

This paper presents a new proof for the Cartan-Brauer-Hua Theorem.
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.
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Cartan-Brauer-Hua Theorem is a well known theorem which states that if R is a subdivision ring of a division ring D which is invariant under all elements of D or dRd􀀀1 � R for all d 2 D n f0g, then either R = D or R is contained in the center of D. The invariance idea of this basic theorem is the main notion of this paper. We prove that if D is a division ring with involution � and M is a subspace of D which is invariant under all symmetric elements of D, then either M is contained in the center of D or is a Lie ideal of D. Also, we show that if T is a self-invariant subfield of a non-commutative division ring D with a non-trivial automorphism, then T contains at least one non-central proper subfield of D.
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In this paper a duality principle is formulated for statements about skew field extensions of finite (left or right) degree. A proof for this duality principle is given by constructing for every extension L/K of finite degree a dual extension LJK, . These dual extensions are constructed by embedding a given L/K in an inner Galois extension N/K. The Appendix shows that such an embedding can always be constructed, and introduces the notion of an inner closure N for L/K. In Section 1 some properties of inner Galois (or bicentral) extensions are mentioned, leading to the notion of a dual extension and the duality theorem. In Section 2 it is shown that the basic structures of LJK can be described by those of a dual extension L,/K,. A survey of this is the translation table at the end of Section 2. Based on these translations the general duality principle is formulated. In Section 3 we establish dual connections between a number of notions known in the literature. For instance, it appears that cyclic Galois exten- sions and binomial extensions are duals of each other. In Section 4 dual connections are used to prove in an easy way some results on cyclic Galois extensions, generalizing Amitsur’s results [l]. Compared to [S], in this paper in Section 1 the proofs are much shorter and generalized to the new notion of a predual extension. In Sections 2 and 4 derivations are also handled. Section 3 follows some parts of Chapter 7 of [S]. In Section 4 most of the material is new, although some special cases already were handled in [S]. The construction in the Appendix is new. We continue with some basic terminology. By a field we mean a skew field; we denote fields by K, L, N, D, E with or without subscripts. If Kc L