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281

Geometry & Topology Monographs Volume 3: Invitation to higher local fields Part II, section 8, pages 281­292

8. Higher local skew fields
Alexander Zheglov

n -dimensional local skew fields are a natural generalization of n -dimensional local fields. The latter have numerous applications to problems of algebraic geometry, both arithmetical and geometrical, as it is shown in this volume. From this viewpoint, it would be reasonable to restrict oneself to commutative fields only. Nevertheless, already in class field theory one meets non-commutative rings which are skew fields finite-dimensional over their center K . For example, K is a (commutative) local field and the skew field represents elements of the Brauer group of the field K (see also an example below). In [ Pa ] A.N. Parshin pointed out another class of non-commutative local fields arising in differential equations and showed that these skew fields possess many features of commutative fields. He defined a skew field of formal pseudodifferential operators in n variables and studied some of their properties. He raised a problem of classifying non-commutative local skew fields. In this section we treat the case of n = 2 and list a number of results, in particular a classification of certain types of 2-dimensional local skew fields.

8.1. Basic definitions
Definition. A skew field complete with respect to commutative). A field K fields K = Kn , Kn-1 , . valuation skew field with
K is called a complete discrete valuation skew field if K is a discrete valuation (the residue skew field is not necessarily is called an n -dimensional local skew field if there are skew . . , K0 such that each Ki for i > 0 is a complete discrete residue skew field Ki-1 .

Examples. (1) Let k be a field. Formal pseudo-differential operators over k ((X )) form a 2- dimensional local skew field K = k ((X ))((X1 )), X X = X X + 1 . If char (k ) = 0 we get an example of a skew field which is an infinite dimensional vector space over its centre.
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(2) Let L be a local field of equal characteristic (of any dimension). Then an element of Br(L) is an example of a skew field which is finite dimensional over its centre.

From now on let K be MK2 and t be a generato 1 system of local parameters valuations of K2 and K1

a two-dimensional r of MK1 . If t1 of K . We denote associated with t2

local skew field. Let t2 be a generator of K is a lifting of t then t1 , t2 is called a 1 by vK2 and vK1 the (surjective) discrete a n d t1 .

Definition. A two-dimensional local skew field K is said to split if there is a section of the homomorphism OK2 K1 where OK2 is the ring of integers of K2 . Example (N. Dubrovin). Let Q ((u)) x, y be a free associative algebra over Q ((u)) with generators x, y . Let I = [x, [x, y ]], [y , [x, y ]] . Then the quotient
A = Q ((u)) x, y /I

is a Q -algebra which has no non-trivial zero divisors, and in which z = [x, y ] + I is a central element. Any element of A can be uniquely represented in the form
f0 + f1 z + . . . + fm z
m

where f0 , . . . , fm are polynomials in the variables x, y . One can define a discrete valuation w on A such that w(x) = w([x, y ]) = 1 , w(a) = k if a = fk z k + . . . + fm z m , fk = 0 . fractions of A has a discrete valuation v which is a unique completion of B with respect to v is a two-dimensional local not split (for details see [ Zh, Lemma 9 ]). Definition. Assume that K1 is a field. The homomorphism
0 : K Int(K ), 0 (x)(y ) = x-1 y x

w(y ) = w(Q ((u))) = 0 , The skew field B of extension of w . The skew field which does

induces a homomorphism : K2 /O 2 Aut(K1 ). The canonical automorphism of K K1 is = (t2 ) where t2 is an arbitrary prime element of K2 .

Definition. Two two-dimensional local skew fields K and K are isomorphic if there is an isomorphism K K which maps OK onto OK , MK onto MK and OK1 onto OK1 , MK1 onto MK1 .

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8.2. Canonical automorphisms of infinite order
Theorem. (1) Let K be a two-dimensional local skew field. If n = id for all n 1 then char (K2 ) = char (K1 ), K splits and K is isomorphic to a two-dimensional local skew field K1 ((t2 )) where t2 a = (a)t2 for all a K1 . (2) Let K, K be two-dimensional local skew fields and let K1 , K1 be fields. Let n 1 . Then K is isomorphic to K if and only if n = id , = id for all n there is an isomorphism f : K1 K1 such that = f -1 f where , are the canonical automorphisms of K1 and K1 . Remarks. 1. This theorem is true for any higher local skew field. 2. There are examples (similar to Dubrovin's example) of local skew fields which do not split and in which n = id for some positive integer n . Proof. (2) follows from (1). We sketch the proof of (1). For details see [ Zh, Th.1 ]. If char (K ) = char (K1 ) then char (K1 ) = p > 0 . Hence v(p) = r > 0 . Then for any element t K with v(t) = 0 we have ptp-1 r (t) mod MK where t is the image of t in K1 . But on the other hand, pt = tp , a contradiction. Let F be the prime field in K . Since char (K ) = char (K1 ) the field F is a subring of O = OK2 . One can easily show that there exists an element c K1 such that n (c) = c for every n 1 [ Zh, Lemma 5 ]. Then any lifting c in O of c is transcendental over F . Hence we can embed the field F (c ) in O . Let L be a maximal field extension of F (c ) which can be embedded in O . Denote by L its image in O . Take a K1 \ L . We claim that there exists a lifting a O of a such that a commutes with every element in L . To prove this fact we use the completeness of O in the following argument. Take any lifting a in O of a. For every element x L we have axa-1 x mod MK . If t2 is a prime element of K2 we can write
axa-1 = x + 1 (x)t
2

where 1 (x) O . The map 1 : L x 1 (x) K1 is an -derivation, i.e.
1 (ef ) = 1 (e)(f ) + e1 (f )

for all e, f L . Take an element h such that (h) = h , then 1 (a) = g (a) - ag where g = 1 (h)/((h) - h). Therefore there is a1 K1 such that
(1 + a1 t2 )axa-1 (1 + a1 t2 )-1 x mod M2 . K

By induction we can find an element a = . . . · (1 + a1 t2 )a such that a xa

-1

= x.

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Now, if a is not algebraic over L , then for its lifting a O which commutes with L we would deduce that L(a ) is a field extension of F (c ) which can be embedded in O , which contradicts the maximality of L . Hence a is algebraic and separable over L . Using a generalization of Hensel's Lemma [ Zh, Prop.4 ] we can find a lifting a of a such that a commutes with elements of L and a is algebraic over L , which again leads to a contradiction. k Finally let a be purely inseparable over L , ap = x , x L . Let a be its lifting pk which commutes with every element of L . Then a - x commutes with every pk element of L . If vK (a - x) = r = then similarly to the beginning of this proof pk pk we deduce that the image of (a - x)c(a - x)-1 in K1 is equal to r (c) (which pk is distinct from c ), a contradiction. Therefore, a = x and the field L(a ) is a field extension of F (c ) which can be embedded in O , which contradicts the maximality of L . Thus, L = K1 . To prove that K is isomorphic to a skew field K1 ((t2 )) where t2 a = (a)t2 one can apply similar arguments as in the proof of the existence of an element a such that -1 a xa = x (see above). So, one can find a parameter t2 with a given property. In some cases we have a complete classification of local skew fields. Proposition ([ Zh ]). Assume that K1 is isomorphic to k ((t1 )). Put
= (t1 )t
-1

1

mod M

K1

.

Put i = 1 if is not a root of unity in k and i = vK1 (n (t1 ) - t1 ) if is a primitive n th root. Assume that k is of characteristic zero. Then there is an automorphism f Autk (K1 ) such that f -1 f = where
(t1 ) = t1 + xti + x2 y t 1
2i -1 1

for some x k /k (i -1) , y k . Two automorphisms and are conjugate if and only if
( (), i , x(), y ()) = ( ( ), i , x( ), y ( )).

Proof. First we prove that = f f

-1

where
2in+1 1

i (t1 ) = t1 + xt1n+1 + y t

for some natural i. Then we prove that i = i . Consider a set {i : i N } where i = fi i xi k , 1 = . Write

-1 -1 fi

, fi (t1 ) = t1 + xi ti for some 1

i (t1 ) = t1 + a2,i t2 + a3,i t3 + . . . . 1 1
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One can check that a2,2 = x2 ( 2 - ) + a2,1 and hence there exists an element x2 such that a2,2 = 0 . Since aj,i+1 = aj,i , we have a2,j = 0 for all j 2 . Fu r a3,3 = x3 ( 3 - ) + a3,2 and hence there exists an element x3 k such that a3,3 Then a3,j = 0 for all j 3 . Thus, any element ak,k can be made equal to ze n |(k - 1), and therefore = f f -1 where ~
(t1 ) = t1 + ai ~ ~
in+1 n+1 t1

k ther, = 0. ro if

+ ai ~

in+n+1 n+n+1 t1

+ ...
n+1

for some i, aj k . Notice that ain+1 does not depend on xi . Put x = x() = ai ~ ~ ~ Now we replace by . One can check that if n|(k - 1) then ~
aj,k = aj,
k -1

.

for 2

j < k + in

a nd
ak
+in,k

= xk x(k - in - 1) + ak

+in

+ some polynomial which does not depend on xk .

From this fact it immediately follows that a2in+1,in+1 does not depend on xi and for all k = in + 1 ak+in,k can be made equal to zero. Then y = y () = a2in+1,in+1 . Now we prove that i = i . Using the formula
(t1 ) = t1 + nx()
n -1 in+1

t

1

+ ...

we get i = in + 1 . Then one can check that vK1 (f -1 (n - id )f ) = vK1 (n - id ) = i . n Since - id = f -1 (n - id )f , we get the identity i = i . The rest of the proof is clear. For details see [ Zh, Lemma 6 and Prop.5 ].

8.3. Canonical automorphisms of finite order
8.3.1. Characteristic zero case. Assume that a two-dimensional local skew field K splits, K1 is a field, K0 Z (K ), char (K ) = char (K0 ) = 0 , n = id for some n 1 , for any convergent sequence (aj ) in K1 the sequence (t2 aj t

2

-1

) converges in K .

Lemma. K is isomorphic to a two-dimensional local skew field K1 ((t2 )) where
t2 at-1 = (a) + i (a)ti + 2i (a)t2i + 2 2 2
2i+n

(a)t

2i+n 2

+ ...

for all a K

1

where n|i and j : K1 K1 are linear maps and
i (ab) = i (a)(b) + (a)i (b)

for every a, b K1 .

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Moreover
2 tn at-n = a + i (a)ti + 2i (a)t2i + 22 2

2i+n



(a)t

where j are linear maps and i and := 2i - ((i + 1)/2)

2i+n 2 2
i

+ ...

are derivations.

Remark. The following fact holds for the field K of any characteristic: K is isomorphic to a two-dimensional local skew field K1 ((t2 )) where
i t2 at-1 = (a) + i (a)t2 + 2 i+1

(a)t

i+1

2

+ ...

where j are linear maps which satisfy some identity. For explicit formulas see [ Zh, Prop.2 and Cor.1 ]. Proof. It is clear that K is isomorphic to a two-dimensional local skew field K1 ((t2 )) where t2 at-1 = (a) + 1 (a)t2 + 2 (a)t2 + . . . for all a 2 2 and j are linear maps. Then 1 is a (2 , ) -derivation, that is 1 (ab) = 1 (a)2 (b) + (a)1 (b). I nde e d,
t2 abt
2
-1 - = t2 at2 1 t2 bt -1

2

= ((a) + 1 (a)t2 + . . . )((b) + 1 (b)t2 + . . . )

= (a)(b) + (1 (a) (b) + (a)1 (b))t2 + . . . = (ab) + 1 (ab)t2 + . . . .

2

From the proof of Theorem 8.2 it follows that 1 is an inner derivation, i.e. 1 (a) = g 2 (a) - (a)g for some g K1 , and that there exists a t2,2 = (1 + x1 t2 )t2 such that
t
2,2 1 at-2 = (a) + 2, 2,2

(a)t

2 2,2

+ ... .

One can easily check that 2,2 is a (3 , ) -derivation. Then it is an inner derivation and there exists t2,3 such that
t
2,3 1 at-3 = (a) + 2, 3,3

(a)t

3 2,3

+ ... .
kn 2,j j

By induction one deduces that if
t
2,j
-1 at2,j = (a) + n,j

(a)t

n 2,j

+ ... +

k n,j

(a)t

+ j,j (a)t

2,j

+ ...

then
t

j,j

is a (j +1 , ) -derivation and there exists t
n,j

2,j +1

such that
+
j +1,j +1

2,j +1

1 at-j +1 = (a) + 2,

(a)t

n 2,j +1

+ ... +

k n,j

(a)t

kn 2,j +1

(a)t

j +1 2,j +1

+ ... .

The rest of the proof is clear. For details see [ Zh, Prop.2, Cor.1, Lemmas 10, 3 ]. Definition. Let i = vK2 ((tn )(t1 ) - t1 ) nN , ( is defined in subsection 8.1) 2 - n and let r Z/i be vK1 (x) mod i where x is the residue of ((t2 )(t1 ) - t1 )t2 i . Put
a = rest
(2i -
1



i+1 2 2 i (t )2 i1

)(t1 )

dt

1

K0 .

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Part II. Section 8. Higher local skew fields
( i ,

287

2i

are the maps from the preceding lemma).

Proposition. If n = 1 then i, r don't depend on the choice of a system of local parameters; if i = 1 then a does not depend on the choice of a system of local parameters; if n = 1 then a depends only on the maps i+1 , . . . , 2i-1 , i, r depend only on the maps j , j nN , j < i. / Proof. We comment on the statement first. The maps j are uniquely defined by parameters t1 , t2 and they depend on the choice of these parameters. So the claim that i, r depend only on the maps j , j nN , j < i means that i, r don't depend on the / choice of parameters t1 , t2 which preserve the maps j , j nN , j < i. / Note that r depends only on i. Hence it is sufficient to prove the proposition only for i and a. Moreover it suffices to prove it for the case where n = 1 , i = 1 , because / if n = 1 then the sets {j : j nN } and {i+1 : . . . , 2i-1 } are empty. It is clear that i depends on j , j nN . Indeed, it is known that 1 is an inner / (2 , ) -derivation (see the proof of the lemma). By [ Zh, Lemma 3 ] we can change a parameter t2 such that 1 can be made equal 1 (t1 ) = t1 . Then one can see that i = 1 . From the other hand we can change a parameter t2 such that 1 can be made equal to 0. In this case i > 1 . This means that i depends on 1 . By [ Zh, Cor.3 ] any map j is uniquely determined by the maps q , q < j and by an element j (t1 ). Then using similar arguments and induction one deduces that i depends on other maps j , j nN , j < i . / Now we prove that i does not depend on the choice of parameters t1 , t2 which preserve the maps j , j nN , j < i. / Note that i does not depend on the choice of t1 : indeed, if t1 = t1 + bz j , b K1 then z n t z -n = z n t1 z -n + (z n bz -n )z j = t + r , where r Mi \Mi+1 . One can see K K 1 1 that the same is true for t = c1 t1 + c2 t2 + . . . , cj K0 . 2 1 Let q be the first non-zero map for given t1 , t2 . If q = i then by [ Zh, Lemma 8, (ii) ] there exists a parameter t such that z t z -1 = t + q+1 (t )z q+1 + . . . . Using this 1 1 1 1 fact and Proposition 8.2 we can reduce the proof to the case where q = i, (t1 ) = t1 , (i (t1 )) = i (t1 ) (this case is equivalent to the case of n = 1 ). Then we apply [ Zh, Lemma 3 ] to show that
vK2 (((t ) - 1)(t1 )) = vK2 (((t2 ) - 1)(t1 )), 2

for any parameters t2 , t , i.e. i does not depend on the choice of a parameter t2 . For 2 details see [ Zh, Prop.6 ]. To prove that a depends only on i+1 , . . . , 2i-1 we use the fact that for any pair of parameters t , t we can find parameters t = t1 + r , where r Mi , t such that K 2 1 12 corresponding maps j are equal for all j . Then by [ Zh, Lemma 8 ] a does not depend on t and by [ Zh, Lemma 3 ] a depends on t = t2 + a1 t2 + . . . , aj K1 if and only 2 2 1 if a1 = . . . = ai-1 . Using direct calculations one can check that a doesn't depend on t = a0 t2 , a0 K1 . 2
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To prove the fact it is sufficient to prove it for t = t1 + cth z j for any j < i, c K0 . 1 1 Using [ Zh, Lemma 8 ] one can reduce the proof to the assertion that some identity holds. The identity is, in fact, some equation on residue elements. One can check it by direct calculations. For details see [ Zh, Prop.7 ]. Remark. The numbers i, r, a can be defined only for local skew fields which splits. One can check that the definition can not be extended to the skew field in Dubrovin's example. Theorem. (1) K is isomorphic to a two-dimensional local skew field K0 ((t1 ))((t2 )) such that
t2 t1 t
-1

2

= t1 + xti + y t 2

2i 2

r where is a primitive n th root, x = ct1 , c K0 /(K0 )d ,

y = (a + r (i + 1)/2)t

-1 2

1

x,

d = gcd(r - 1, i).

If n = 1 , i = , then K is a field. (2) Let K, K be two-dimensional local skew fields of characteristic zero which splits; n and let K1 , K1 be fields. Let n = id , = id for some n, n 1 . Then K is isomorphic to K if and only if K0 is isomorphic to K0 and the ordered sets (n, , i, r, c, a) and (n , , i , r , c , a ) coincide. Proof. (2) follows from the Proposition of 8.2 and (1). We sketch the proof of (1). From Proposition 8.2 it follows that there exists t1 such that (t1 ) = t1 ; i (t1 ) can be represented as ctr ai . Hence there exists t2 such that 1
t2 t1 t
-1

2

= t1 + xti + 2i (t1 )t2i + . . . 2 2
K

Using [ Zh, Lemma 8 ] we can find a parameter t = t1 mod M 1
t2 t t 1
-1

such that

2

= t1 + xti + y t2i + . . . 2 2

The rest of the proof is similar to the proof of the lemma. Using [ Zh, Lemma 3 ] one can find a parameter t = t2 mod M2 such that j (t1 ) = 0 , j > 2i. K 2 Corollary. Every two-dimensional local skew field K with the ordered set
(n, , i, r, c, a)

is a finite-dimensional extension of a skew field with the ordered set (1, 1, 1, 0, 1, a). Remark. There is a construction of a two-dimensional local skew field with a given set (n, , i, r, c, a).
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Examples. (1) The ring of formal pseudo-differential equations is the skew field with the set (n = 1, = 1, i = 1, r = 0, c = 1, a = 0). (2) The elements of Br(L) where L is a two-dimensional local field of equal characteristic are local skew fields. If, for example, L is a C2 - field, they split and i = . Hence any division algebra in Br(L) is cyclic. 8.3.2. Characteristic p case. Theorem. Suppose that a two-dimensional K0 Z (K ), char (K ) = char (K0 ) = p > 2 Then K is a finite dimensional vector isomorphic to a two-dimensional local skew
t
p with x K1 , (i, p) = 1 . -1

local skew field K splits, K1 is a field, and = id . space over its center if and only if K is field K0 ((t1 ))((t2 )) where

2

t1 t2 = t1 + xti 2

Proof. The "if " part is obvious. We s If K is a finite dimensional vec algebra over a henselian field. In fact, k ((u))((t)). Then by [ JW, Prop.1.7 ] k Hence there exists t1 such that tp 1 as a vector space with the relation
t2 t1 t
-1

ketch the proof of the "only if " part. tor space over its center then K is a division the center of K is a two-dimensional local field K1 /(Z (K ))1 is a purely inseparable extension. Z (K ) for some k N and K K0 ((t1 ))((t2 ))

2

i = t1 + i (t1 )t2 + . . .

(see Remark 8.3.1). Then it is sufficient to show that i is prime to p and there exist parameters t1 K1 , t2 such that the maps j satisfy the following property: (*) If j is not divisible by i then j = 0 . with some cj /i K1 . Indeed, if this property holds then by cj /i = ((i + 1) . . . (i(j /i - 1) + 1))/(j /i)!. b K1 such that j satisfies the same prop
t
-1

If j is divisible by i then j = c

j /i j /i i

induction one deduces that cj /i K0 , Then one can find a parameter t2 = bt2 , 2 = 0 . Then erty and i

2

t1 t = t1 - i (t1 )ti . 2 2

First we prove that (i, p) = 1 . To show it we prove that if p|i then there exists a k map j such that j (tp ) = 0 . To find this map one can use [ Zh, Cor.1 ] to show that 1
ip (tp ) = 1 Then property this map idea. 0, we ( *) we ip2 (tp 1 prove doe s n r e duc e
2

) th ot th

= 0 , . . . , ipk (tp ) = 0 . 1 at for some t2 property (*) holds. To show it we prove that if pk hold then there exists a map j such that j (t1 ) = 0 . To find e proof to the case of i 1 mod p . Then we apply the following

k

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Let j 1 mod p be the minimal positive integer such that j is not equal to zero on
K1 . Then one can prove that the maps m , k j satisfy the following property: there exist elements cm,k K1 such that (m - c
m, 1 p
l

m < (k + 1)j , k {1, . . . , p - 1}

- ... - c
pl

m, k

k )|K

pl

=0

1

where : K1 K1 is a linear map, |K
(t1 ) = 1 , c
p
l

k j,k

= c(j (t1 )) , c K0 .

p

l

k

1

is a derivation, (tj ) = 0 for j pl N , / 1

Now consider maps q which are defined by the following formula
t
-1

2

at2 = a + i (a)ti + 2

i+1

(a)t

i+1

2

+ ... ,

a K1 .

Then q + q + k=1 k q-k = 0 for any q . In fact, q satisfy some identity which is similar to the identity in [ Zh, Cor.1 ]. Using that identity one can deduce that if j 1 mod p and there exists the minimal m ( m Z ) such that mp+2i |K pl = 0 if j |(mp + 2i) and
p
l

q-1

mp+2i |K p 1

l

= s

(2i+mp)/j
j

1

|K

pl

for any s K1 otherwise, and

1

q (t1 ) = 0 for q < mp + 2i, q 1 mod p ,

then
(mp + 2i) + (p - 1)j is the minimal integer such that
(mp+2i)+(p-1)j |K p 1
l+1

= 0.

To complete the proof we use induction and [ Zh, Lemma 3 ] to show that there exist l pl 2 parameters t1 K1 , t2 such that q (tp ) = 0 for q 1, 2 mod p and j = 0 on K1 . 1

Corollary 1. If K is a finite dimensional division algebra over its center then its index is equal to p . Corollary 2. Suppose that a two-dimensional local skew field K splits, K1 is a field, K0 Z (K ), char (K ) = char (K0 ) = p > 2 , K is a finite dimensional division algebra over its center of index pk . Then either K is a cyclic division algebra or has index p . Proof. By [ JW, Prop. 1.7 ] K1 /Z (K ) is the compositum of a purely inseparable extension and a cyclic Galois extension. Then the canonical automorphism has order pl for some l N . By [ Zh, Lemma 10 ] (which is true also for char (K ) = p > 0 ), K K0 ((t1 ))((t2 )) with
- t2 at2 1 = (a) + i (a)ti + 2 i+p
l

(a)t

i+p 2

l

+

i+p

l

(a)t

i+2p 2

l

+ ...

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where i pl N , a K1 . Suppose that = 1 and K1 is not a cyclic extension of Z (K ). Then there exists a field F K1 , F Z (K ) such that |F = 1 . If a F m then for some m the element ap belongs to a cyclic extension of the field Z (K ), m hence j (ap ) = 0 for all j . But we can apply the same arguments as in the proof of the preceding theorem to show that if i = 0 then there exists a map j such that m j (ap ) = 0 , a contradiction. We only need to apply [ Zh, Prop.2 ] instead of [ Zh, Cor.1 ] and note that = x where is a derivation on K1 , x K1 , x 1 mod MK1 , because (t1 )/t1 1 mod MK1 . - Hence t2 at2 1 = (a) and K1 /Z (K ) is a cyclic extension and K is a cyclic division algebra (K1 (tp )/Z (K ), , tp ). 2 2 Corollary 3. Let F = F0 ((t1 ))((t2 )) be a two-dimensional local field, where F0 is an algebraically closed field. Let A be a division algebra over F . Then A B C , where B is a cyclic division algebra of index prime to p and C is either cyclic (as in Corollary 2) or C is a local skew field from the theorem of index p . Proof. Note that F is a C2 -field. Then A1 is a field, A1 /F1 purely inseparable extension and a cyclic Galois extension, and u A1 . Hence A splits. So, A is a splitting two-dimensional It is easy to see that the index of A is |A : F | = pq m , q subalgebras B = CA (F1 ), C = CA (F2 ) where F1 = F (up ), [ M, Th.1 ] A B C . The rest of the proof is clear. Now one can easily deduce that Corollary 4. The following conjecture: the exponent of A is equal to its index for any division algebra A over a C2 -field F (see for example [ PY, 3.4.5. ]) has the positive answer for F = F0 ((t1 ))((t2 )). is the A1 = local (m, p F2 = compositum of a F0 ((u)) for some skew field. ) = 1 . Consider F (um ). Then by
k k

Reference
[JW] [M] [Pa] B. Jacob and A. Wadsworth, Division algebras over Henselian fields, J.Algebra,128(1990), p. 126­179. P. Morandi, Henselisation of a valued division algebra, J. Algebra, 122(1989), p.232­243. A.N. Parshin, On a ring of formal pseudo-differential operators, Proc. Steklov Math. Inst., v.224 , 1999, pp. 266­280, (alg-geom/ 9911098).

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292 [PY] [Zh]

A. Zheglov V.P. Platonov and V.I. Yanchevskii, Finite dimensional division algebras, VINITY, 77(1991), p.144-262 (in Russian) A. B. Zheglov, On the structure of two-dimensional local skew fields, to appear in Izv. RAN Math. (2000).

Department of Algebra, Steklov Institute, Ul. Gubkina, 8, Moscow GSP-1, 117966 Russia. E-mail: abzv24@mail.ru, azheglov@chat.ru

Geometry & Topology Monographs, Volume 3 (2000) ­ Invitation to higher local fields