An Involutorial Version of a Theorem of Gräter

Abstract

Let D be a division ring with centre Z and with involution ( * ). Let V be a valuation of D with value group , a linearly ordered additive group (non necessarily commutative) together with a symbol (positive infinity). We assume that for each nonzero symmetric element s = s * of D, which is algebraic over Z, we have for all nonzero elements x of D, V xa − ax > V ax . We define the residue characteristic exponent p of V to be the characteristic of the associated residue division ring written as D V , if = 0, and p = 1, if = 0. We show here that if F is a finite dimensional commutative subalgebra of D over Z, which is * -closed (i.e., F * = F ), and if ( * ) is of the first kind (i.e., each central element of D must be symmetric), then F Z = 2 r p m where m is a nonnegative integer and r = 0 or 1 according as the restricted involution in F is trivial or not. The case of an involution ( * ) of the second kind (i.e., some central element of D is not symmetric) requires (for this author) a stronger type of valuation, namely, V is a * -valuation, that is to say, for all elements x of D, we have V x * = V x , a condition which readily implies must be Abelian. Here, we can show that for F as in the preceding, F Z = p m , where m is again a nonnegative integer. The preceding results generalize a theorem of Gräter and improve in parts recent theorems of this author in [2]. In the special case p = 2 the results provide a modicum of answers to the questions opened informally in [2] (see concluding paragraph in [2] or here Question 3.2.1). More is to be said in the third and final section of this work.