SK_{1 }

OF GRADED DIVISION ALGEBRAS

23

Remark (Dickson Theorem). One can also see that, with the same assumptions as in Th. A.1, if a, b ∈ E

have the same minimal polynomial h ∈ T [x], then a and b are conjugates. For, h = (x

b)k where

k ∈ T [b][x].

# But then by (III), there exists a conjugate of a, say a^{0}, such that k(a^{0}) =6

0.

Since h(a^{0}) = 0,

by (I) some conjugate of a^{0 }is a root of x

b. (This is also deducible using the graded version of the

# Skolem-Noether theorem, see [HwW_{2}], Prop. 1.6.)

# Appendix B. The Congruence theorem for tame division algebras

For a valued division algebra D, the congruence theorem provides a bridge for relating the reduced Whitehead group of D to the reduced Whitehead group of its residue division algebra. This was used by Platonov [P_{1}] to produce non-trivial examples of SK_{1}(D), by carefully choosing D with a suitable residue division algebra. Keeping the notations of Section 2, Platonov’s congruence theorem states that for a division algebra D with a complete discrete valuation of rank 1, such that Z(D) is separable over F , ( 1 + M D ) ∩ D ( 1 ) ⊆ D 0 . T h i s c r u c i a l t h e o r e m w a s e s t a b l i s h e d w i t h a l e n g t h y a n d r a t h e r c o m p l i c a t e d p r o o in [P_{1}]. In [E], Ershov states that the “same” proof will go through for tame valued division algebras over henselian fields. However, this seems highly problematical, as Platonov’s original proof used properties of maximal orders over discrete valuation rings which have no satisfactory analogues for more general valuation rings. For the case of strongly tame division algebras, i.e., char(F ) [D : F ], a short proof of the congruence theorem was given in [H_{2}] and another (in the case of discrete rank 1 valuations) in [Sus]. In this appendix, we provide a complete proof for the general situation of a tame valued division algebra. f

Theorem B.1 (Congruence Theorem). Let F be a eld with a henselian valuation v, and let D be a tame F - c e n t r a l d i v i s i o n a l g e b r a . T h e n ( 1 + M D ) ∩ D ( 1 ) ⊆ D 0 .

Tameness is meant here as described in §2, which is the weaker sense used in [JW] and [E].

Among the

several characterizations of tameness mentioned in §2, the ones we use here are that D is tame if and only if D is split by the maximal tamely ramified extension of F , if and only if char(F ) = 0 or char(F ) = p =6 0 and the p- primary component of D is inertially split, i.e., split by the maximal unramified extension of F .

# The proof of the theorem will use the following well-known lemma:

Lemma B.2. Let D be a division ring with center F and let L be a eld extension of F with [L : F ] = `. If a ∈ D and a ⊗ 1 ∈ (D ⊗_{F }L)^{0}, then a^{` }∈ D^{0}.

P r o o f . T h e r e g u l a r r e p r e s e n t a t i o n L → M ` ( F ) y i e l d s a r i n g m o n o m o r p h i s m D ⊗ F L → M ` ( D ) . T h e r e f o r we have a composition of group homomorphisms e ,

(D ⊗_{F }L)^{∗ }→ GL_{`}(D) → D^{∗}/D^{0},

a 7→

a 0 . . . 0 0 a . . . 0 . . . . . . . . . . . . 0 0 . . . a !

`×`

7→ a^{`}D^{0},

where the second map is the Dieudonne´ determinant. (See [D], determinant.) The lemma follows at once, since the image of the contains (D ⊗_{F }L)^{0}.

§20 for properties of the Dieudonne´ composition is abelian, so its kernel

Note that in the preceding lemma, there is no valuation present, over F .

and D could be of infinite dimension

Proof of Theorem B.1. The proof is carried out in four steps.

Step 1. The theorem is true if D is strongly tame over F , i.e., proof given in [H_{2}] and another (in the case of discrete valuation of convenience of the reader, we recall the argument from [H_{2}]:

char(F )

rank

1)

in

# [D : F ]. This has

[Sus],

Lemma

1.6.

a short For the