PURELY INSEPARABLE, MODULAR EXTENSIONS

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 176, February

1973

PURELY INSEPARABLE, MODULAREXTENSIONS OF UNBOUNDEDEXPONENT BY

LINDA ALMGREN KIME ABSTRACT. Let K be a purely inseparable extension of a field k of characteristic p * 0. Sweedler has shown in [2, p. 4031 that if K over k is of finite exponent, then 7v is modular over k if and only if K can be written as the tensor product of simple extensions of k. This paper grew out of an attempt to find an analogue to this theorem if K is of unbounded exponent over k. The definition of a simple extension is extended to include extensions of the form k [x, xl/i>,

x^-IP

, • • •][%1/P°Û].

If K is the

sions, then K is modular. The converse, counterexamples in §4 illustrate. Even

is still

tensor

product

of simple

exten-

however, is not true, as several if we restrict [k : r ] < oo, the converse

shown to be false.

Given

7Í over

k modular,

we construct

a field \\c_

kKp

always imbeds in K where M is the tensor product of simple old sense. In general K * Q. For K to be the tensor product

sions, we need K = Q, andf)°°

kKpl = k(C\°°_,Kpl).

®M (= Q) that

extensions of simple

in the exten-

If for some finite ,V, kKpN

ftN + 1

= kKp , then we have (by Theorem 11) that K = Q. This finiteness condition guarantees that M is of finite exponent. Should ff?, kKp = k, then we would have the condition of Sweedler's original theorem. The counterexamples in §4 will hopefully be useful to others interested in unbounded exponent extensions. Of more general interest are two side theorems on modularity. These state that any purely inseparable field extension has a unique minimal modular closure, and that the intersection of modular extensions is again modular.

1. Modular rable

extensions

in a common

are said

their

intersection.

minimal unbounded

All fields

of a common

algebraically

fields

ate linearly

extensions.

disjoint

positive

field

extension

such

exponent

throughout

under field

that over

are taken

Received by the editors February AMS (MOS) subject classifications

consideration

k. They

(e.g.

disjoint,

for all positive

number

or infinite

tion and unions

closed

to be linearly

A field

ground

will be purely

are to be viewed

the algebraic

we mean that

K of k is said

closure

i. K is of exponent Kp

C k. If no such

k. Unless

of k).

they are linearly

to be modular

n over ra exists,

otherwise

insepa-

as contained If two

disjoint

if Kp

over

and

k

k if ra is the then

indicated,

K is of

the intersec-

from i (or ra) = 1 to °°. 14, 1972. (1970). Primary 12F15-

Key words and phrases. Purely inseparable, modular, unbounded exponent, tensor product of simple extensions.

modular

closure,

infinite

or

Copyright © 1973, American Mathematical Society

335 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

336

L. A. KIME Lemma

there

exists

1.

Let

an

Mna.-, £ \Mn] y a.' such

and

Q are linearly

\xv

• ■• , xn]C

Proof.

Given

the linear

Ma

{Mj

then

any set

|x,,•«•,

£ \Ma].

If ixp

disjointness

of Ma

fx,,...,

of fields that

disjoint,

M

n Q. Hence

be a set

such

both

KJaMa

[February that,

M„a,

and

for any

and

M„a2

0 are linearly

x ¡ C Ua Ma, there

and

0,

\x

an a.

dependent

over

such

over

. ■. , x [ is linearly

dependent

£ \Mj, M„a

disjoint.

exists

• • - , x ¡ is linearly

x f is linear

Ma.,, Ma

lie in M„clt . If' each

that

0, then

dependent

by

over

\JaMaC\

Q (D Ma

Ma is modular

over

n Q).

Q.E.D. Lemma Ua

^a

2.

Let

zs m°dulaT

Proof.

over

« > 1. Given

\xv

, xs]C

it is linearly

tees

that

any

that

(|Ja*la)i'

(x., ••.,

independent

..,

over

positive

(= Ua

xj

is linearly

exists

over

over

k «=> k

p

over

that

k n (\J a Mpa"),

of M

k.

disjoint

an aQ such

independent

independent

K modular

disjoint

over

over

k guaran-

Q.E.D.

O K is modular

k. We want to show that

for all positive

over

k for all

linearly

independent

k, \x .,•••,

over

Kp" n k = (k

x ] is linearly

Case 2. « < z. (*">" /fep! "n

K''1, (^^^"n

Conversely,

\J(kl/p"

has

k oí finite

p

shown

exponent

L is modular

now extend

this

Theorem (1)

p

There

4.

n K is modular

over

Kp , hence

over

linearly

xj

C /fe is

over k ' p

of K Ci Kp .

is contained

in a unique

k. L is called to the infinite

K be a purely a unique

exponent

minimal

field

inseparable field

closure

L is purely

inseparable

(3)

If K is of exponent

over «, then

extension extension

of K over

7< L,

k. We

case. field

extension

extension

of k.

L D K where

k. (2)

«, then

Q.E.D.

minimal

the modular

inseparable

dis joint.

k for any positive

in [2, p. 4081 that any purely

over

exists

and

n K)"'' = kpi~" n K^'. Since kpÍ~" O Kp¿ n zfe=

definition Let

O K)p

n Kp ) n k, by the modularity

independent

n K) = K is modular over ¿ by Lemma 2.

Sweedler

where

O Kp!. If jxp...,

7C)i' and k are trivially

if k

(k /p

i.

Case 1. n > i. (k1/p" n K)p' = kl/p"~'

over

) anc^ ^ are linearly

k O Mpa . The modularity

K is modular

Assume

linearly

over

k, then

n.

Proof.

over

^a

x ] C (Ja Mpa there

\xx, • • •, x ] is linearly

3.

1. If each

k.

/M£o". if {xv.

Proposition

k ate

be as in Lemma

We must show

for all

...

\Ma]

k. L has exponent

« over

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k.

L is modular

1973]

PURELY INSEPARABLE, MODULAR EXTENSIONS

Proof. has

Since

kl/p'

a corresponding

O K has exponent

modular

closure

L.

337

p* over k, by [2, p. 408] kup'

satisfying

(1)—(3).

n K

The minimality

of

each L . guarantees that L. CL, CL, C ••• . By Lemma 2, \JL . = L is modular over k. L clearly contains K, since kl/p' Í1KCL. and K = \J(kUpt n K). Any field

extension

of K that

minimality

of each

L .). Thus

Naturally

L is called

Corollary

5.

Let

ground

field

a common Proof.

Let

is modular

Ç\aMa

Lemma.6, any positive

such

that

\xPn

is a p-basis

of Mae/l

for field extensions

for kl/p"

O Kp over

n A/

k1/pl n *K» = *[{*,!, íx2¡, •••, «*fn

= it [(A n **)[{*, |,..., I'roposition

over

7.

7/ K over

A for any positive Proof.

Let

iy^,

A /s zz modular

\x.\] = k[kUpl n Kp\. Q.E.D. extension

then

kKp

is modular

ra. ly2!, •■-,!)'•+[!

be a modular

basis

for A

1 ' ti '+ I p n K over

A, i.e., kl/pi +l O K = A[!yj|l ® A[!y2!l® • •• ® A[{y,+1ll where, for each y. £ |y.|,

y. has exponent

/ over

A. Then

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338

L. A. KIME

[February

kl/pi C\kKp = k[kUpi n Kp] (by Lemma 6)

= *tU1/í,í+1n

K)p] = k[\yp],...,\yp+l]]

= k[\yp]]®...®k[\yp+A] => k

p Cl k Kp is modular

=» (by Proposition

Now replacing hence

in general Lemma

tive

8.

3) that

over

k Kp is modular

K by kKp, we have that that

kKp"

is modular

If K is relatively

k(kKp)p

over

perfect

& for any

¿.

over

over

k.

= /fezK^2is modular

over k,

Q.E.D. k, i.e.

K = kKp - kKp'

i, and K is modular over k, then k1/p O K = kl/p

/or a«y positive

z > 0,

for all posi-

n kKp" = k[kl/p

n K»"]

n. .

Proof.

k[k1/pl

Since

K is modular

over k, we have from Lemma 6 that

k^^pl

n kKp =

n Kp] for any i > 0. Using this and the fact that K = kKp" fot any «>0,

we have that

kvp n K = /fe17i,n ak^aU17»

n k»1

= k[(kp)up2 n a»k»21 = A . *'[(**)"^ Applying

this argument

repeatedly

we get that

kl/p

n a»V31 = k[k1/p n K"2]. n K = kl/p

n £K?* =

AU^On K*"]. Q.E.D. Proposition fect

subfield

9.

If K over

(\ kKp

k is a modular

is modular

over

extension

k. If \x\

then the relatively

is a p-basis

for k

p

perH

(0 kKpi) over k1/p" n (fï AKP¿), then \xpn] is a p-basis for kl/p n (flAK»*) oz,er k. Proof.

Apply Corollary

plicity denote C\kKpl

5 and Proposition

7 to obtain

modularity.

For sim-

by L. The modularity of L over k =» ixp 1 is p-indepen-

dent over A.

A17? n L = kUp n 4L*" = AtA17" n L»"l

(by Lemma 6)

..AU1'*"*1 n Ll»" = AftA17»"n DM»" = A(AnL»")[lx»"|] = A[{x»"l]. We note that a relatively is not necessarily

Let

modular,

perfect

as the following

k = Z (x», y», z» ), x, y, z

\w,w1/p,wl/p2,..,\

field

byW1/p°°.

extension example

= II ^iC

shows.

indeterminates.

Set K = *[*l""\

Q.E.D.

K over k (K = kK

Denote

any

(** + y)1""].

set

41/2 C K

over /fe has diagram z, xz + y

(cf. [2, p. 4021) -* fc17»2 n K over k is not a modularextension. 3, K is not modular

over

k.

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By Proposition

)

19731

PURELY INSEPARABLE, MODULAREXTENSIONS

Lemma A and

A

10.

If A, B are field

p O A, B are linearly

extensions disjoint

of k, such

over

k, then

that

339

A is modular

A, B are linearly

over

disjoint

over k.

Proof.

If ix i is a p-basis

p-independent independent the linear

over (kl/pn over

O A)(b).

of A

of finite

Otherwise

11.

products

Aray modular

N, kKp' = kKp of K of finite

Proof.

of the

n A), which

over (A O Ap )(BP

x . is a basis

is p-

) C B, contradicting

for A over

for A • B over

A consisting B =* A ®, B

over A. Q.E.D.

field

extension

K over

k, where

for some finite

exponent.

Let

p O K over

choice

(C kl/p

is

, is isomorphic to (I AKP ® M where M is a modular subfield

ixjj

be a p-basis

\xpA is p-independent/A. A

\xp"\

p D A and B. So the basis

~ AB =» A, B are linearly disjoint Theorem

O A over A17p" n A, then jxj

A, would be p-dependent

disjointness

of monomials

for Al7p*

A. A

is not unique,

of [x |, since

(which equals

kl,p

for kX/p

Ci K over

Let A , be a completion

A.

but the choice

is precisely

kl/p

n K. By modularity

of {x^j to a p-basis

a p-basis

of A.

for A

for

is independent

pn

K over

k[k

of the

p C\ Kp\

n AKP by Lemma 6). Hence Al7p n K = (A17" n AKP) ®

A[Aj] => AKP® A[Aj1 c-, k (by Lemma 10). Let over

A

\x2\

be a p-basis

p n K. Choose

for A17p A

n K over

as any completion

over Al7p O K, i.e., a p-basis

kl/p

O K. jx"j

of \xp\

is p-independent

to a p-basis

for A

p (~) K

for kUp2 O K over (A1/p n K)(A17p2 n Kp).

A[ixf }1= AU17" nKp2l = Al7pn kKp2,so kupn

K = (A17" n kKp2) ® k[Ap]

® A[A,1 =»AKp2® A[A21® A[Aj] c_ k. In

general,

we

let

A

be

a p-basis

for

A

p

H K

over

(¿i/i>*- 1 n K)(A17p*n Kp)" Al7pnK= (Al7pn AKP")® A[Af " 'l ® • • • ® AU/! -» AKP"® A[AJ ® • ■• ® A[A J c» K. So for any finite ra, AKP" and A[Aj U •• • U A 1 are linearly disjoint

over A => fl AKP' and A[Aj u • • • U A\

ate linearly disjoint over A. By Lemma 1, (J A[Aj U ... ij AJ = k[A ^ A^J.-A and H AKP' are linearly disjoint over A, i.e., D AKP'® A[Aj u A2 u • • •] imbeds in K. By assumption,

there

exists

a finite

TV such

ftN

AKP

ftN + 1

= AKP

, so A . = 0

tot

all i > N. Set Al= A[Aj U . . . u Avl. M is modular since A[A j u • • • U AN] =

A[A1]®A[A21® ... ®A[AN1 (cf. §3). We assert that K= f|AKp'® M. Let \xv ■.. , xs\

be a p-basis

A1/pn K= A[xp¿, ...,

for A1/p'+1

xp!l®A[A£/V-1l®

p-independent over A. t[xf',•••, (M AKP ) (since

i > N).

n K over Al7p! n K, where

Proposition

...®

z > N. Then

k[AA\, where *»',...,

*J* are

x*'l = A[A17p n Kp'] = Al7p O AKP*= A17p n 9 says

that we may assume

CkUpi +1n (f|AKp¿). Q.E.D. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

that {*,, ••.,

xs\

340

L. A. KIME Remark

12.

Either

condition

that

[A

[February

p O K: k] < os or [A : kp] < oo implies

that, for some N, kKpN = kKpN +l. Since if [kUp n K : k] < o. (note [k:kp] implies that [k1/p O K : k] < oo), then ¿N = 0 With only the restriction

that

K over A is modular

I 1 kKp ® A [A j u A2 u • • •! imbeds empty

A .'s, A [A j ij A2 u • •.]

describe

in K. Since

K/k,

a purely

finite exponent if, for any field has finite

exponent

A[A j u A intermediate

over

may be infinitely

of finite

finite

have that

exponent.

many non-

We can

exponent.

extension,

is said

L, A C L C K, where [k

to be of locally

p n L : k] < oo, then L

A.

u • • •] is easily

field

of locally

inseparable

we always

there

is not, in general,

k[A x u A2 u • . •] as being

Definition.

kKpN = kKpN + \

L, such

seen

that

[k

to be of locally

finite

p C\ L : k] < oo, there 2

such that LCA[A1uA2u-,-U^

uAp + .u

exponent.

must exist

Given any a finite

«

Ap +. . . .] => L is of exponent

n

over A.

Though we have C\kKp' ® A[A . U A2 U •.• *=1k[x\/pu])

as counterexamples

restrict

product

Q.E.D.

(b) and (c) show

If we

and

Kp

by the linear

extensions,

K may still

that

over Qp" n A, and hence

It is not true,

examples

k[xla/p°°])

«. Q is modular

and fe are linearly disjoint. simple

K Ot (®a

extensions.

A, then,

linearly dependent

to the tensor

A.

*f| C K, J*,,....

5, r and

(in the old sense)

dent

over

341

exponent).

M is the tensor

the following

structure

product theorem

] then (1 AKP is the tensor product of simple

K will be the tensor

Given

Hence

is modular,

a field

product

extension

of simple

K over

extensions.

k, if A[f|Kp

1 is modular,

then

A[flKp!] S* aA[zi7p00lfor some {zJ C C\KP'. Proof. \zla/p"\

Let \zJ C f|Kp'

must

be a p-basis for AI/pn

be p-independent

over

A17p"o

A[OKp!],

p-basis for Al7p" + 1 n A[f|Kpi]

over Al7p"n

A[f|Kp!]

was not a p-basis

would imply that \za\

A[fÏKp!l

over A. Then

If \zla/p"\

A[fÏKp'l, for k

were

not a

then the modularity of p C\ A[IIKP

] over A.

Hence AI7p"n A[flKp'l » ®ak[z)¿pn~ M- A[flKpîl * ®ak[z\/p°°]. Q.E.D. Theorem of simple

16.

Given

extensions

K over

A a modular

extension,

K is the tensor

of A if and only if (in the terminology

product

of Theorem

11) K =

ÓaKp¿® A[A,u A2 U---1 and f| AKP¿= A[D Kp']. Proof.

A[flKp'l

"-»".

"«=".

The remarks

® A[Aj U Au

Assume that

preceding

•. .], then

Theorem

K is the tensor

K S «8>ak[xla/p°°])

15 showed

product

® (^^Aty^l).

that if K S

of simple

extensions.

Since each yß has

finite exponent over A, A[DKp!l C flAKp! s a^^Í/p°°] c ^[riKp,l,

riAKpî'=AtflK"'].

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i.e.

342

r

L. A. KIME By Proposition

[February

13 we have that

k= n(r\kKpi®k[A1u = /Dak^'saUjU

a2u...i)

a2u...1)

((®A[x^n) ® (k[yJ\y (®k[xvp°°]\

= fi A/C»'® A[Aj U A2 u ...1 i (since ®aA[x^»~] C A[f|¿ Kpl] C f\ Ail*1).Q.E.D. We have

[k1/pn

shown

(Remark

12) that

if Afv^

= AT^'

for some finite

K:k] « => AT^7 = A(f\ (2)

Kp ){x ,,--•,

(a) The following

of unbounded

In particular

K over A (i.e.

be the prime field

exponent

counterexample

we will construct

K = C\kKp'), of characteristic

a modular

p, and x.,

Let A = Z ÂX y X2, Xj, • • •)

kl/p2

Ä^"n

relatively

product

perfect

where K ¿ k[f\Kpl].

indeterminates.

A^n

illustrates

need not be a tensor

K=k[x\/p] O K = A[x^2x^]

K=A[x^"x^"-1...x^l

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x2, x^,.

• • be

that of

1973]

343

PURELY INSEPARABLE, MODULAR EXTENSIONS

It is easy to verify that K is of the form ÍI AKP . We claim that K 4 A[f|Kp in particular

that

MKP

Assume

not.

Then there

1,

C A. exists

a y £ ÍIKP

where

y £ A - Ap (set difference).

Then y will be in all of the following fields: A n KP = Z (x,, xp, xp,...) p i i y

fl(An

A nKp2 =Zp(xlXP, xp, xf,

xp2,...)

k nKp* =ZAx,x* ft 1 2 ...xp'-\ i

*f1

Kpl)CZp(xv

x\, xp2,...),soy ,+1

finite

eZ^x,, P

y £ ZAx,

ra. y £ A O KP"T1

^■1A2

(= G).. We claim that FnG y eFn

exte nsion

the following ô

1

, • • • ,xp

2

„«-1

*"- 1 „

A

n

1

n~KP" ■*„ n + 1 ,P"1 2 ,n + l

our hypothesis

diagram

of Z. (xp , xp

in — 1

xp,

= Zp(xpxp¿ ... xp", xpn, xf

GC Ap, which contradicts

Consider

, xpl, 2 xp',...) 3 '

G using

is Z (xp.

an(^ restricted

be a diagram

for

the above ,n + l

the ground

diagram.

Recall j« + I n + 1 ). to Z (xp , xp ,n + l

field

F.

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that the

If we deleted , ... ,xp

),

344

L. A. KIME The

((x{x*

elements

of G ate

those

•••*Pnn~l)>**n)i

in the

[February

span

of monomials

and ((x,x»...xf-V)'

in

where

i = 0,. . ., p - 1,

j4o,...,p»-l-i. Hence

F H G can be described

coefficients

in

Z (x^ p

FOG

(b)

, x%

1

as polynomials . . .

¿

= Z (xpxp2 ...xp", p

The following

l

2

argument

[A :kp] < oo, a modular

xp

)

with

'

xpn, xp"+1,..,,xpn

n

• • • xp

), i.e.

n

12'

shows

extension

in (x,x^

that

even

may not always

with

'

+ l). Q.E.D.

r?

^-

the restriction

be a tensor

that

product

of simple

extensions.

Let

A be any field

tion is not perfect,

A well-known K over

fields.

where

|A| is infinite.

from Galois

18.

when

fields

x.z— ,]. 1

A

19.

There

Proof.

with z.

Trivial,

are at most

|A| = N0, and

given

over

any field

number

exten-

of intermediate

A and of exponent

one,

p.

Ci L) - A, there exists

and y are p-independent

earlier

that

Q.E.D.

Theorem

x.

that,

p n K = A[x,,1' x„2' • •.,

modular by [2, p. 4031. By Proposition Lemma

Assume

D 7 p. A by assump-

are a finite

extension

7< is of the form A[x.,

over A and x»z e A[x.,..., 1'

states

inseparable

inseparable

K : kl/p"

Any such

theory

A there

of intermediate

Any purely

[k1/p" + lD

over

K is purely

number

that

of A.

theorem

In the case

Proof.

in particular

closure

A, 7< is primitive

there are a finite Lemma

p / 0 such

hence

let L be the perfect

sion

of characteristic

of A[xj].

there

A[x.,

over

y])

Since

are an infinite have

A (since

exponent

TV,an a such that TV= (Afx^Hal

[k:kp]

a y £ (kl/p > p).

Mp C k[x A and number 2 over

= A[a].

n L) - A such that

Consider

Mj = A[xj

p, y]

[M : k[x A] = p , by an

of intermediate A. We associate

fields, with

all of which each

such

field

For any given a, [k[a] : k] = p2, and

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1973]

PURELY INSEPARABLE, MODULAR EXTENSIONS

if a 4 a!, number

k[x.,

a] = A[a)

of candidates

Having of A[xj,

chosen

Each

of these

= A[xj,

a].

Clearly

the

a's

provide an infinite

for an x

an x

we consider

%21. Mp2C k[xj,

ber of intermediate

4 k[d\

345

x2],

fields

provides

M, = A[xj,

[M2 :A[xj,

of degree a candidate

x2'p,

y] as a field

x2]] = p , so there

p over

A[x

for x

such

xA that

exist

extension

an infinite

and exponent

3 over

if x. 4 *?, then

num-

A.

A[x.,x2, x,]

4 A[xj, x2, x'J. We proceed exist

in the above

an infinite

Hence

there

number

exist

manner

of choices N0

at least

NQ

to choose such K0

=2

that,

an x ,, x_, etc.

choices

for modular

field

of the form f|AKpz, where [A1/p n K : A] = p and [A:AP] > p. (c)

In the following

extension

example,

For each

x. there

if x.lit4 x'., A[x,,- • •, x.]i 4 k[x,,... \

we will explicitly

construct

extensions

, x'J. i K/A

Q.E.D. a modular

field

K over A, where K = f| AKP'^ A[fl Kp*] and [A : Ap] <

ua

We also have that x +9£A17pr'+2n

K

n +2

= Z ft(x17p* +2fl}7p" +1fl,17p".,.aI7pV7¿ 12 n UV «2''"' Consider

the following

fl +T, ö +-,,•••)

n+2

over

n + 3'

diagram

°

°

*„+2' flf,+3' fln+4'--°

(=G)-

of Z(xl/pn+2,

a!7p"+1, fli7p",. • •, fl1/p2, fl17?,

a......

flp,-,,

Z (x, a,,

p

n+1

1

p

,fl

2'

'

1

...

n +1

2

n +2

'

'

n

n +1

flp.,,•••).

n + 3'

,'V

,! '(,"+'

,iV

uiV2ai/»-+I

n+1

(,}'»

,'Vy.-1

fl'i>"

The above (containing

would

fl¿7p,

G consists

/=0,...,p"

(e) a tensor

n + 1'

for F if we deleted

of monomials

in ((x ' p

assumption final

complement

n+2'

n + 3'

the far right-hand

a 'p"

and (U1/pn+2«J/p" +1---«yp2)V

+ I-l.

This

,>/»

)"

box

aPn+y...).

of the span

an + v aPn+2, fl^23,...). our initial

'»

be for a diagram

fln+2,

a„+2' K+3)¿'-"

(a

,^"

So F n G = Zp((x^pn+2a\pn [F nG:A] that

[A[x

counterexample over

shows

A in fl AKP

We let A = Z (x, y, z v z2,..

+ l ...

al/p2)p,

av a2,. .:,

*1

+ 1vl/T>n + 1

r "7 Z l/pn+2

diagram

overZ(x,y,z^-,...,z^-,2n

1

>, z}/Pn+2,...,zl/pn

« + 1-7

_l/z> rz+2

l/pn+2

Jfr1".y.*

v

. y

+ 2, zl/p)

r?

l/i)"+1

' zl

:-z+l

l/i)n+2

(=C). l/i)"+2

'""s2

l/fi\

. z +?)

+ 1).

.1 />"..! 'P

. P

„1 0"

(x'

_« +2

F consists which

(in terms

of polynomials

(ZV*)'

(j JO,...,

2I-'>" + 1vl

of the

Z (x, y, Zj

of the above

in l(xl/pn+2

(l = 0,...,p-l)

.1 P yl V)i> n + I

p" + !

p

modular

, • • • , z„ ? basis)

+ z1/pn + :yl/P"

no power

) span

,!

,1 P

of monomials

of z 'M occurs.

+ 1 + .. . + zl/p2y

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in

G consists

Up2) + *$y1/»li,

and (x17>"+2 + zYpn +1y1/pn"+1 + ...+

p" +1 - 1). Hence

P

zypn2y^'P2)P-i

19731

PURELY INSEPARABLE, MODULAR EXTENSIONS

F O G-Z

p

((x;/p-"+2_z!7p"

+ 1y17p" + 1 + ...

349

+z!/p2v1/p2)p, v

_l/p"+2

,l/p"+2)

Then *p"^

£AU;/P,..,,

contradicts

our

n+2

1

z17p]=,x.

£A[z}/P,.,.,

'nil

z1/pj^x, n

1

£A,

r

which

assumption

that

[A(xj)

:A] = p.

BIBLIOGRAPHY 1.

theory, 2.

N. Jacobson,

Van Nostrand, M. Sweedler,

401-410.

Lectures

Princeton, Structure

in abstract

algebra.

N. J., 1964. of inseparable

Vol.

3: Theory

of fields

and Galois

MR 30 #3087. extensions,

Ann. of Math. (2.) 87 (1968),

MR 36 #6391.

DEPARTMENT OF MATHEMATICS, BOSTON STATE COLLEGE, BOSTON, MASSACHUSETTS 02115

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