TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 176, February 1973
PURELY INSEPARABLE, MODULAR EXTENSIONS
OF UNBOUNDED EXPONENT
BY
LINDA ALMGREN KIME
ABSTRACT. Let K be a purely inseparable extension of a field k of char-
acteristic 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 tensor product of simple exten-
sions, then K is modular. The converse, however, is not true, as several
counterexamples in §4 illustrate. Even if we restrict [k : r ] < oo, the converse
is still shown to be false.
Given 7Í over k modular, we construct a field \\c_ kKp ®M (= Q) that
always imbeds in K where M is the tensor product of simple extensions in the
old sense. In general K * Q. For K to be the tensor product of simple exten-
sions, we need K = Q, andf)°° kKpl = k(C\°°_,Kpl). 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 extensions. All fields under consideration will be purely insepa-
rable extensions of a common ground field k. They are to be viewed as contained
in a common algebraically closed field (e.g. the algebraic closure of k). If two
fields are said to be linearly disjoint, we mean that they are linearly disjoint over
their intersection. A field extension K of k is said to be modular if Kp and k
ate linearly disjoint for all positive i. K is of exponent n over k if ra is the
minimal positive number such that Kp C k. If no such ra exists, then K is of
unbounded or infinite exponent over k. Unless otherwise indicated, the intersec-
tion and unions throughout are taken from i (or ra) = 1 to °°.
Received by the editors February 14, 1972.
AMS (MOS) subject classifications (1970). Primary 12F15-
Key words and phrases. Purely inseparable, modular, modular closure, infinite or
unbounded exponent, tensor product of simple extensions.
Copyright © 1973, American Mathematical Society
335
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336 L. A. KIME [February
Lemma 1. Let {Mj be a set of fields such that, for any Ma.,, Ma £ \Mj,
there exists an Mn £ \Mn] such that both M„ and M„ lie in M„ . If each M„a.-, y a.' a, a2 clt ' a
and Q are linearly disjoint, then KJaMa and 0 are linearly disjoint.
Proof. Given any set |x,,•«•, x ¡ C Ua Ma, there exists an a. such that
\xv • ■ • , xn]C M £ \Ma]. If ixp • • - , x ¡ is linearly dependent over 0, then by
the linear disjointness of Ma and 0, \x . ■ . , x [ is linearly dependent over
Ma n Q. Hence fx,,..., x f is linear dependent over \JaMaC\ Q (D Ma n Q).
Q.E.D.
Lemma 2. Let \Ma] be as in Lemma 1. If each Ma is modular over k, then
Ua ^a zs m°dulaT over k.
Proof. We must show that (|Ja*la)i' (= Ua ^a ) anc^ ^ are linearly disjoint
for all « > 1. Given any (x., ••., x ] C (Ja Mpa there exists an aQ such that
\xv ... , xs]C /M£o". if {xv. .., xj is linearly independent over k n (\J a Mpa"),
it is linearly independent over k O Mpa . The modularity of M over k guaran-
tees that \xx, • • •, x ] is linearly independent over k. Q.E.D.
Proposition 3. K is modular over k «=> k p O K is modular over k for all
positive n.
Proof. Assume K modular over k. We want to show that (k /p O K)p and
k ate linearly disjoint for all positive i.
Case 1. n > i. (k1/p" n K)p' = kl/p"~' O Kp!. If jxp..., xj C /fe is
linearly independent over Kp" n k = (k p n Kp ) n k, by the modularity of K
over k, \x .,•••, x ] is linearly independent over Kp , hence over k ' p Ci Kp .
Case 2. « < z. (*">" n K)"'' = kpi~" n K^'. Since kpÍ~" O Kp¿ n zfe =
/fep! "n K''1, (^^^"n 7C)i' and k are trivially linearly dis joint.
Conversely, if k p n K is modular over k for any positive «, then
\J(kl/p" n K) = K is modular over ¿ by Lemma 2. Q.E.D.
Sweedler has shown in [2, p. 4081 that any purely inseparable extension 7<
over k oí finite exponent is contained in a unique minimal field extension L,
where L is modular over k. L is called the modular closure of K over k. We
now extend this definition to the infinite exponent case.
Theorem 4. Let K be a purely inseparable field extension of k.
(1) There exists a unique minimal field extension L D K where L is modular
over k.
(2) L is purely inseparable over k.
(3) If K is of exponent «, then L has exponent « over k.
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1973] PURELY INSEPARABLE, MODULAR EXTENSIONS 337
Proof. Since kl/p' O K has exponent p* over k, by [2, p. 408] kup' n K
has a corresponding modular closure L. satisfying (1)—(3). 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 is modular over k must contain all the 7. . (by the
minimality of each L .). Thus L must be the unique minimal such field. Q.E.D.
Naturally L is called the modular closure of 7< over k.
Corollary 5. Let \Ma\ be a set of fields such that each Ma is modular over
a common ground field k. Then Ç\aMa is modular over k.
Proof. Let N be the modular closure of Mae/l Ma. By minimality, N C Ma
for all a e A. Hence NCf)aMa. Q.E.D.
2. A structure theorem for field extensions of unbounded exponent.
Lemma.6, If K over k is modular, then k(kl/p' n Kp) = k)/p' O k.Kp for
any positive i.
Proof. K = \J(kupi O K)-K'=UU1/<,IVl Kp)=*kKp =\Jk(kx/pi C\ Kp).
If \xn\ is a p-basis for kl/p" O Kp over A1/p"~ ' Pi Kp, \xj is ^-indepen-
dent over k1/pn~l n A/<p <=» S*£"_1! is p-independent over An kp"~l Kp".
\xp"~ j (C (k p n Kp") C Kp") is by assumption p-independent over
U1'^*" : O KO""" '^nK^.soby the linear disjointness of k and Kp",
\xPn | is p-independent over k, hence p-independent over kC\kf Kp . So \x \ is p-
independent over kl/p"~lnkKp. We also have that \xp\c(knKp)[\xl\,\x2\,. ■ ■ ,\xn__l\]
C ^Lix,!,---,ixn_]il where each ¡x.( isa p-basis for k1/p'r\Kp over A17*'"1 O Kp.
AK* - \Jk(kUp' n K*) - = A[|x,l, |x2¡,.. ]
where each |x.| £ A1^' o AK" is p-independent over kU'p'~ O kKp,
\xp\ c k[\x x\, \x2\,..., ix;._,n
=»each ix.i is a p-basis for A1''"' n AK" over kUpl~l C\ kK"
=>k1/pl n *K» = *[{*,!, íx2¡, •••, «*fn
= it [(A n **)[{*, |,..., \x.\] = k[kUpl n Kp\. Q.E.D.
I'roposition 7. 7/ K over A /s zz modular extension then kKp is modular
over A for any positive ra.
1 ' ti '+ IProof. Let iy^, ly2!, •■-,!)'•+[! be a modular basis for A 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 over & for any z > 0,
=» (by Proposition 3) that k Kp is modular over k.
Now replacing K by kKp, we have that k(kKp)p = /fezK^2 is modular over k,
hence in general that kKp" is modular over ¿. Q.E.D.
Lemma 8. If K is relatively perfect over k, i.e. K = kKp - kKp' for all posi-
tive i, and K is modular over k, then k1/p O K = kl/p n kKp" = k[kl/p n K»"]
/or a«y positive n.
.
Proof. Since K is modular over k, we have from Lemma 6 that k^^pl n kKp =
k[k1/pl 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 . *'[(**)"^ n a»V31 = k[k1/p n K"2].
Applying this argument repeatedly we get that kl/p n K = kl/p n £K?* =
AU^On K*"]. Q.E.D.
Proposition 9. If K over k is a modular extension then the relatively per-
fect subfield (\ kKp is modular over k. If \x\ is a p-basis for k p H
(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 5 and Proposition 7 to obtain modularity. For sim-
plicity denote C\kKpl 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]. Q.E.D.
We note that a relatively perfect field extension K over k (K = kK = II ^iC )
is not necessarily modular, as the following example shows.
Let k = Z (x», y», z» ), x, y, z indeterminates. Denote any set
\w,w1/p,wl/p2,..,\ byW1/p°°. Set K = *[*l""\ (** + y)1""]. 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. By Proposition
3, K is not modular over k.
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19731 PURELY INSEPARABLE, MODULAR EXTENSIONS 339
Lemma 10. If A, B are field extensions of k, such that A is modular over
A and A p O A, B are linearly disjoint over k, then A, B are linearly disjoint
over k.
Proof. If ix i is a p-basis for Al7p* O A over A17p" n A, then jxj is
p-independent over (kl/pn O A)(b). Otherwise \xp"\ (C kl/p n A), which is p-
independent over A, would be p-dependent over (A O Ap )(BP ) C B, contradicting
the linear disjointness of A p D A and B. So the basis for A over A consisting
of monomials of finite products of the x . is a basis for A • B over B =* A ®, B
~ AB =» A, B are linearly disjoint over A. Q.E.D.
Theorem 11. Aray modular field extension K over k, where for some finite
N, kKp' = kKp , is isomorphic to (I AKP ® M where M is a modular subfield
of K of finite exponent.
Proof. Let ixjj be a p-basis for kX/p Ci K over kl/p n K. By modularity
\xpA is p-independent/A. Let A , be a completion of {x^j to a p-basis for
A p O K over A. A is not unique, but the choice of A. is independent of the
choice of [x |, since A. is precisely a p-basis for A pn K over k[k p C\ Kp\
(which equals kl,p n AKP by Lemma 6). Hence Al7p n K = (A17" n AKP) ®
A[Aj] => AKP® A[Aj1 c-, k (by Lemma 10).
Let \x2\ be a p-basis for A17p n K over kl/p O K. jx"j is p-independent
over A p n K. Choose A as any completion of \xp\ to a p-basis for A p (~) K
over Al7p O K, i.e., a p-basis 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 [A J ® • ■ • ® 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 A J = 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.ftN ftN + 1
By assumption, there exists a finite TV such AKP = AKP , so A . = 0 tot
all i > N. Set Al = A[Aj U . . . u Avl. M is modular since A [A j u • • • U A N] =
A[A1]®A[A21® ... ®A[AN1 (cf. §3). We assert that K= f|AKp'® M. Let
\xv ■.. , xs\ be a p-basis for A1/p'+1 n K over Al7p! n K, where z > N. Then
A1/pn K= A[xp¿, ..., xp!l®A[A£/V-1l® ...® k[AA\, where *»',..., *J* are
p-independent over A. t[xf',•••, x*'l = A[A17p n Kp'] = Al7p O AKP* = A17p n
(M AKP ) (since i > N). Proposition 9 says that we may assume that {*,, ••., xs\
CkUpi + 1n (f|AKp¿). Q.E.D.
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340 L. A. KIME [February
Remark 12. Either condition that [A 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] < œ
implies that [k1/p O K : k] < oo), then ¿N = 0 for some /V => kKpN = kKpN + \
With only the restriction that K over A is modular we always have that
I 1 kKp ® A [A j u A2 u • • •! imbeds in K. Since there may be infinitely many non-
empty A .'s, A [A j ij A2 u • •.] is not, in general, of finite exponent. We can
describe k[A x u A2 u • . •] as being of locally finite exponent.
Definition. K/k, a purely inseparable extension, is said to be of locally
finite exponent if, for any field L, A C L C K, where [k p n L : k] < oo, then L
has finite exponent over A.
A[A j u A u • • •] is easily seen to be of locally finite exponent. Given any
intermediate field L, such that [k p C\ L : k] < oo, there must exist a finite «2
such that LCA[A1uA2u-,-U^ uAp + .u Ap +. . . .] => L is of exponent n
over A.
Though we have C\kKp' ® A [A . U A2 U •.• <] C-» K, counterexample (d) in §4
shows that we need not have (~\ kKp' ® A[A j u A2 U . . .] = K. We do have the
following:
Proposition 13. Given K over k modular, set Q= (\ kKp ® k[A .u Aaj...].
Then K is modular and relatively perfect over Q (i.e., K = (~)QKP ).
Proof. Let \xj be a p-basis for kl/p" O Q over k1/p" l n Q. \xj is p-
independent over A p n K (since Q is a modular subfield of 7<). If \yn] is
a completion of {x | toa p-basis for A p C\ K over A p OK, then iy^S is
a possible choice for iy„_1l for all positive « and hence \y j | u iy2| u íy?i u •••
is a modular basis for K over Q. Q.E.D.
Counterexample (e) in §4 shows that Theorem 11 cannot be refined by
"tensoring off" k[C\ Kp'] from f]kKp'.
3. A generalization of simple extensions and a tensor product theorem.
Sweedler has shown in [2, p. 403] that any purely inseparable extension K over
A of finite exponent is modular if and only if it is isomorphic to the tensor pro-
duct (over A) of simple extensions of A. In this section we will study the problem
as to whether this theorem has an analogue in the infinite exponent case. We must
first extend our definition of a simple extension to include those of the form
k[x,xp,x p ,•••]= A [x p ]. We may now ask whether a modular extension
of infinite exponent is modular if and only if it is isomorphic to the tensor product
of simple extensions of A.
The following theorem gives us the implication in one direction.
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1973] PURELY INSEPARABLE, MODULAR EXTENSIONS 341
Theorem 14. If K is isomorphic to the tensor product of simple extensions
(of A), then K is modular over A.
Proof. By assumption K Ot (®a k[xla/p°°]) ® (®ßk[yß]). To prove that K
is modular we need to show that Kp and A are linearly disjoint for all positive
ra. Given any \zv ..., *f| C K, J*,,.... *,! C [«g>*=1 k[x\/pu]) ® (®J=,*[yy])](=ß)
for some finite 5, r and «. Q is modular since it is the tensor product of simple
(in the old sense) extensions. Hence if \zp , . . . , zp j C Kp is linearly depen-
dent over A, then, by the linear disjointness of Qp and A, \zp , • •., zp j is
linearly dependent over Qp" n A, and hence over Kp" D A D ÖP" n A). Thus Kp"
and fe are linearly disjoint. Q.E.D.
It is not true, however, that if K is modular, that it is the tensor product of
simple extensions, as counterexamples (a) and (e) of §4 demonstrate. Counter-
examples (b) and (c) show that even with the added restriction that [A : Ap] < t»,
K may still not be the tensor product of simple extensions.
If we restrict K over A such that AKP = AKP for some N
(or [A17p n K:A] < « or [A :kp] < <*>) by Theorem 11 we know that K » flAKp'
® M, where M is modular (and of finite exponent). Hence M is the tensor product
of simple extensions. Since f IAKP is modular, the following structure theorem
shows that if (l AKP = A[flKp ] then (1 AKP is the tensor product of simple
extensions. Thus K will be the tensor product of simple extensions.
Theorem 15. Given a field extension K over k, if A[f|Kp 1 is modular, then
A[flKp!] S* <g>aA[zi7p00l for some {zJ C C\KP'.
Proof. Let \zJ C f|Kp' be a p-basis for AI/pn A[fÏKp!l over A. Then
\zla/p"\ must be p-independent over A17p"o A[OKp!], If \zla/p"\ were not a
p-basis for Al7p" + 1 n A[f|Kpi] over Al7p"n A[fÏKp'l, then the modularity of
A[f|Kp!] would imply that \za\ was not a p-basis for k 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 16. Given K over A a modular extension, K is the tensor product
of simple extensions of A if and only if (in the terminology of Theorem 11) K =
ÓaKp¿® A[A,u A2 U---1 and f| AKP¿ = A[D Kp'].
Proof. "«=". The remarks preceding Theorem 15 showed that if K S
A[flKp'l ® A[Aj U Au •. .], then K is the tensor product of simple extensions.
"-»". Assume that K S «8>ak[xla/p°°]) ® (^^Aty^l). Since each yß has
finite exponent over A, A[DKp!l C flAKp! s <8>a^^Í/p°°] c ^[riKp,l, i.e.
riAKpî'= AtflK"'].
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r342 L. A. KIME [February
By Proposition 13 we have that
k= n(r\kKpi®k[A1u a2u...i) ((®A[x^n) ® (<s>k[yJ\y
= /Dak^'saUjU a2u...1) (®k[xvp°°]\
= fi A/C»'® A[Aj U A2 u ...1i
(since ®aA[x^»~] C A[f|¿ Kpl] C f\ Ail*1). Q.E.D.
We have shown (Remark 12) that if Afv^ = AT^' for some finite TV, or
[k1/pn K:k] <oo, or [A: A»] < oo, then we will have K^C\kKp' ® k[AxU A2U...].
The following theorem gives some conditions under which M A7<^ = k[f\Kp ].
Theorem 17. Let K over k be a modular extension.
(1) If K over Ç\Kpl is finitely generated as a field, then DkKp' = AtflK^l.
(2) If there exists a generating set for K over Kp that generates K over
C\Kpi, then C\kKpi = A[flKpi].
Proof. (1) Assume K = (f|. Kp ){x ,,--•, x ). There exists a finite n such
that xpn £ k, i=l,..., m. Hence Kpl = (f| - Kp') (xpl ,.. ., xp!) C k(f) ■ Kp') for
any / > « => AT^7 = A(f\ Kpl) for any ; > «. Therefore f| . vfe/C^* = A(f|¿ K»').
(2) Assume that \x] is a generating set for K over 7<p such that K =
(C\{ Kpi)(\x]). For any positive «, kKp" = Atifl,■ Kpi) i\xp"]]]. Hence if as
nBAK»", a EOjkKCii Kpi)[{xp"]]]) = kin, Kpi]. Q.E.D.
4. Some counterexamples, (a) The following counterexample illustrates that
a modular field extension of unbounded exponent need not be a tensor product of
simple extensions. In particular we will construct a modular relatively perfect
field extension K over A (i.e. K = C\kKp'), where K ¿ k[f\Kpl].
Let Z be the prime field of characteristic p, and x., x2, x^,. • • be
indeterminates.
Let
A = Z ÂX y X2, Xj, • • •)
A^n K=k[x\/p]
kl/p2 O K = A[x^2x^]
Ä^"n K=A[x^"x^"-1...x^l
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1973] PURELY INSEPARABLE, MODULAR EXTENSIONS 343
It is easy to verify that K is of the form ÍI AKP . We claim that K 4 A[f|Kp 1,
in particular that MKP C A.
Assume not. Then there 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
A nKp2 =Zp(xlXP, xp, xf, xp2,...)
k nKp* =ZAx,x* ...xp'-\ *f , xpl, xp',...)ft 1 2 i 1 2 3 '
fl(An Kpl)CZp(xv x\, xp2,...),soy eZ^x,, xp,in — 1
, + 1finite ra. y £ A O KP"T1 ..t>n~KP" ,P"y £ ZAx, P
^■1A2
v" X ., X , , X_■*„ n + 1 1 2
) (= F) for some
n + 1 .. x"" + 1)' n + 1 '
,n + l(= G).. We claim that FnG = Zp(xpxp¿ ... xp", xpn, xf
y eFn GC Ap, which contradicts our hypothesis that y £ A - Ap.
,n + l). So
Consider the following diagram for ZAx., xp ,.,n-\
exte nsion of Z. (xp , xp , • • • ,xpô 1 2
n + 1 i + l
Cn + 1 ).
XP" ) as a fieldn + 1
(X1X2
(X¡xp
„«-1). x-e
*"- 1 „ A<p )p.xp
n 1
We want to describe F and G using the above diagram. Recall that the,n + l ,n + l j« + I
n + 1). If we deletedground field for this diagram is Z (xp.
xn + l ^rom tne diagram an(^ restricted the ground field to Z (xp , xp , ... ,xp ),
the above would be a diagram for F.
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344 L. A. KIME [February
The elements of G ate those in the span of monomials in
((x{x* •••*Pnn~l)>**n)i and ((x,x»...xf-V)' where i = 0,. . ., p - 1,
j4o,...,p»-l-i.
Hence F H G can be described as polynomials in (x,x^ • • • xp ) with
coefficients in Z (x^ , x% . . . xp ), i.e.p 1 ¿ n '
FOG = Z (xpxp2 ...xp", xpn, xp"+1,..,,xpn + l). Q.E.D.p l 2 n 12' ' r? ^-
(b) The following argument shows that even with the restriction that
[A :kp] < oo, a modular extension may not always be a tensor product of simple
extensions.
Let A be any field of characteristic p / 0 such that [A : kp] > p. A by assump-
tion is not perfect, hence in particular |A| is infinite. Assume that |A| = N0, and
let L be the perfect closure of A.
A well-known theorem from Galois theory states that, given any field exten-
sion K over A, 7< is primitive over A <=> there are a finite number of intermediate
fields. In the case when K is purely inseparable over A and of exponent one,
there are a finite number of intermediate fields <=» [K : k] = p.
Lemma 18. Any purely inseparable extension K over A of the form (1 kKp ,
where [k1/p" + lD K : kl/p" D 7<] = p, is modular.
Proof. Any such 7< is of the form A[x., x2 ,• • •] where x . has exponent i
over A and x» e A[x.,..., x. ,]. A p n K = A[x,, x„ • •., x 1 = A[x ] isz 1' ' z— 1 1' 2' n n
modular by [2, p. 4031. By Proposition 3, Tí is modular over A. Q.E.D.
Lemma 19. There are at most |A|.= N0 purely inseparable modular extensions
K of the form k[C\Kp'], where [k1/p O K : k] = p (i.e., of the form k[z1/p°°]).
Proof. Trivial, since we may associate any field of the form A[z p ](z£k)
with z. Q.E.D.
Theorem 20. There exist 2 purely inseparable modular field extensions of
the form O W, where [k1/p n K : k] = p and [k:kp] > p.
Proof. The field extensions under consideration are of the form
Atxj, x x,...] where [A[x¿] : Al = pl and xp £ A[xp .. . , x._ A.
Given any Xj £(kl/p Ci L) - A, there exists a y £ (kl/p n L) - A such that
x. and y are p-independent over A (since [k:kp] > p). Consider Mj = A[xj p, y]
as a field extension of A[xj]. Since Mp C k[x A and [M : k[x A] = p , by an
earlier remark there are an infinite number of intermediate fields, all of which
(excepting A[x., y]) have exponent 2 over A. We associate with each such field
TV, an a such that TV = (Afx^Hal = A[a]. For any given a, [k[a] : k] = p2, and
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1973] PURELY INSEPARABLE, MODULAR EXTENSIONS 345
if a 4 a!, k[x., a] = A[a) 4 k[d\ = A[xj, a]. Clearly the a's provide an infinite
number of candidates for an x
Having chosen an x we consider M, = A[xj, x2'p, y] as a field extension
of A[xj, %21. Mp2 C k[xj, x2], [M2 :A[xj, x2]] = p , so there exist an infinite num-
ber of intermediate fields of degree p over A[x xA and exponent 3 over A.
Each of these provides a candidate for x such that if x. 4 *?, then A[x.,x2, x,]
4 A[xj, x2, x'J.
We proceed in the above manner to choose an x ,, x_, etc. For each x. there
exist an infinite number of choices such that, if x. 4 x'., A[x,,- • •, x.] 4 k[x,,... , x'J.N0 K0 lit i \ i
Hence there exist at least NQ =2 choices for modular field extensions K/A
of the form f|AKpz, where [A1/p n K : A] = p and [A:AP] > p. Q.E.D.
(c) In the following example, we will explicitly construct a modular field
extension K over A, where K = f| AKP'^ A[fl Kp*] and [A : Ap] < <*,.
Let A = Z (x, y, Zj , z2 i • • •) where x, y, z,, z2, ■ . • are indetermi-
nates (note [A :AP1 = p ).
LetA1/p n K = A[xl7p]
k^p2n K = k[xup2 + ziyup]
A1/p3 nK = A[xI/p3 + z;7pyl7p2 + 22y17P]
A17p"n K = A[xl7pn + z;7p"-2yl7p"-1+... + Zn_1y1/p]
Assume that K = A[(lKp ]. Then there exists an a e k - kp such that a e
il Kp . a will lie in all of the following fields.
AnKp = Zp(x, yp, z\/p°°, z¥p°°,...)
A HKp2 = Z (x + zf yp, yp2, z}7*8*, z*7*0*, ...)
A n Kp¿ = Z (x + zp2yp + ... + zpi ,yp!_ \ yp\ z!/p~' ^1/p06 2l/p°
There exists a finite ra such that a £ Z (x, y, zj7p°°,. .. ,zl/p°°) (= F).
But a also £ A O Kp"+2 = Z (x + zp2yp + ... + zp"+2ypn+1 ypn+2 zl7p°° z17p°° ...)ft I ' n+1 ' ' ' ' i '2 '
(=G).
Consider the following diagram for Z (x, y, z!7p°°, z27p°°, . . .) as a fieldpx~' " "1 ' ~2
7I/p°° 7ft"+3 ,
Pv" " ' "1 '••• 'Zn ' Zn + I ' 2n+2 '"''extension of Z>p" + \ yp"+2, z |7*°",.. .', zI7p°°, zpn^,zx/p
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346 L. A. KIME [February
"rz+1
rP"n + 1
„n+2
x-, zP yp
(x ~zp~yp
.." + ' *r,
'■l yp
yP )P
We would like to use the above diagram to describe F and G. If we deleted
the left-hand column (i.e. all pth roots of zp +. ) and restricted the ground field
to Z (xp , yp • zi > • " • > z ) the above would be a diagram for F.
G consists of the span of monomials in {((z^ + 1 ) ) ¡,
l(x + zÇ2y* + .. . + zPnn+lyP") + (zf+j2y^+1)]7' and ((x + zfy* + •• • + z»"+V»*
where z, /' = 0, • • • , p - 1, A = 0, • • •, p" - 1. Hence F O G can be described as
polynomials in (x + zp yp + • ■ • + zp yp ) with coefficients in
Zp(xpn + \ ypn+2, *}/»", • • • , z*70, i.e., F O C = Zp((x + zp2yp + ... + *»"">")»
Un ,«+2 ,/(,o. _ _ 2l/í,o. c ¿P ^ a e ¿P. Contradiction.* ' 1 n
(à) The next counterexample shows (using the terminology of Theorem 11)
that in general for a modular extension K over A, (] kKp ® M /= K.2 3
Set A = Z (x, ap, ap , ap , • • •) where x, a «-,... are indeterminates. Set
k1/p n K = A[x17^, «j, a», fl^2,...]
A1/p2n 7< = A[x1/i'2iz|/^ 8„ «„ 4, a»2,...]1 1" 2 3 4 '
A17"3 n^Atx17^}7"2^7", fl[, fl2- ay 4 <
7< is modular since
k'/pn n K= k[x^p\\/pn-1 ■■■ ̂ _pv av a2,..., an, apn + v ap2+2,...]
= Atx'^V/P""1 • • • a^fA ® Atfll ® A[a2l ® • • • ® A[anl ® A[a^ + 1l ® • • •
We will show that il kKp = A. Using the terminology of Theorem 11, we set
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1973] PURELY INSEPARABLE, MODULAR EXTENSIONS 347
A.= \a.\. Hence we will have (f|AKp') ® A [A x u A2 u •••] = k[ay a ...] ¿ K.
Since it takes only one element to complete \av ap, ap , . ..\ to a p-basis
for A p Pi K over A, and since fl AKP is modular (by Proposition 9), if
MAKP j£ k, then |)AKP = A[xj, x2,. ..] where x{ has exponent i over A, and
x? £A[xj,..., x._jJ. Consider K as lying in the perfect closure L of A. %j £
A[x /p, a,, ap, • ■ • , apn~ ] for some finite ra, so1 2' ' n
xn,2£(kl/pn + 1n L)[X'/pn'2,a)/pn + \aYpn,...,ayp2}
= zypnn, aypn+\ ayp,...,ayp2, «n+1, a£+2, «&,•••> ua
We also have that
x +9£A17pr'+2n Kn + 2
= Z (x17p* + 2fl}7p" + 1fl,17p".,.aI7pV7¿ft 12 n n+1
UV «2''"' *„+2' flf,+3' fln+4'--° (=G)-
Consider the following diagram of Z(xl/pn+2, a!7p"+1, fli7p",. • •, fl1/p2, fl17?,° ° p 1 2 ' ' n n + 1
fl +T, ö +-,,•••) over Z (x, a,, a...... ,fl ... flp,-,, flp.,,•••).n+2 n + 3' p 1 2' ' n + 1 n + 2 n + 3'
,'V
uiV2ai/»-+I ,iV ,! '(,"+'
(,}'»n+1
,^"
fl'i>" ,'Vy.-1 (a '» )" ,>/»n + 1' n+2' n + 3'
The above would be for a diagram for F if we deleted the far right-hand box
(containing fl¿7p, fln+2, aPn+y...).
G consists of the span of monomials in ((x ' p a 'p" • • ■ a 'p ) . a^'p)1,
a„+2' K+3)¿'-" and (U1/pn+2«J/p" + 1---«yp2)V where i = 0,".. , p - 1 and
/=0,...,p" + I-l. So F n G = Zp((x^pn+2a\pn + l ... al/p2)p, av a2,. .:,
an + v aPn+2, fl^23,...). [F nG:A] = p" + 1 => [k[xn+2]:k] <p" + 1, with contradicts
our initial assumption that [A[x + ] :A] = p" 2
(e) This final counterexample shows that in general A[||KP ] does not have
a tensor complement over A in fl AKP (K over A a modular extension).
We let A = Z (x, y, z v z2,.. •) where x, y, zy z2,... are indeterminates.
Let
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348 L. A. KIME [Febru! ■y
kup n K = k[\zyp], x1/p]
kup2 n K = k[{zyp2], x1/p2 + z\/pyl/p]
kl/pi n K = A[íz;17*3¡, x17"3 + 2J7i>2y17"2 + zl'Py1'*}
Clearly A[íz!7p°°S] C k[C\Kp']. We assert that A [{z^^f] has no tensor
complement in DA/<p . This will imply that A[jz.7p ]] = AtflTv*' ], and in partic-
ular be another counterexample showing that in general |) kKp / k[(]Kp ].
Assume that there exists an M C 7< such that M ®, Atiz:7'' ¡] = K. Sincetz J
it takes only one element to complete \z /p ] to a p-basis for A17^" O K over
k p O K, M must be of the form k[xx, x2, • •-1 where x. has exponent i
over A, and xp. £ A [x ... x. ,].z 1' ' z— i
xx £Z(xl/p, y, z\/p,. .. , zlJp) for some finite « =» xn+, 6
z.(. .1/ö"+2 1/ö"+1 ,l/p"+2 zl7""+2) (=F). ^ut
x ., 6 Al7^" + 2 O K = Z (x17»"+2 + zYP" + ly"p!: + l + • •.77 "• 7 O 1
+ zJfr1". y. *l/pn+2 _l/z>1 > *1
rz+2
x . eZ()77+2 Ö
l/p"+2 ,_ -l/i>n + 1vl/T>n + 1+ z
+ z17?v17*>, >, z}/Pn+2,...,zl/pn + 2, zl/p) (=C).« + 1-7 J 1 r? :-z+l
/~ -J i_ í m J- r "7 Z l/pn+2 l/i)"+1 l/i)n+2 l/i)"+2 l/fi\Consider the following diagram of L\x v . y ' zl '""s2 . z +?)
l/t."+2overZ(x,y,z^-,...,z^-,2n + 1).
l/pn+2
. P
(x'_« +2 2I-'>" + 1vl p" + !
.1 />"..! 'P
„1 0"
.1 P yl V)i> n + I,1 P ,! P
F consists of the Z (x, y, Zj p , • • • , z„ ? ) span of monomials in
which (in terms of the above modular basis) no power of z 'M occurs. G consists
of polynomials in l(xl/pn+2 + z1/pn + :yl/P" + 1 + .. . + zl/p2y Up2) + *$y1/»li,
(ZV*)' (l = 0,...,p-l) and (x17>"+2 + zYpn + 1y1/pn"+1 + ...+ zypn2y^'P2)P-i
(j JO,..., p" + 1 - 1). Hence
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19731 PURELY INSEPARABLE, MODULAR EXTENSIONS 349
F O G-Z ((x;/p-"+2_z!7p" + 1y17p" + 1 + ... +z!/p2v1/p2)p,p
v _l/p"+2 ,l/p"+2)
Then
*p"^ £AU;/P,..,, z17p]=,x. £A[z}/P,.,., z1/pj^x, £A,n+2 1 'nil n 1
r
which contradicts our assumption that [A(xj) :A] = p.
BIBLIOGRAPHY
1. N. Jacobson, Lectures in abstract algebra. Vol. 3: Theory of fields and Galois
theory, Van Nostrand, Princeton, N. J., 1964. MR 30 #3087.
2. M. Sweedler, Structure of inseparable extensions, Ann. of Math. (2.) 87 (1968),
401-410. MR 36 #6391.
DEPARTMENT OF MATHEMATICS, BOSTON STATE COLLEGE, BOSTON, MASSACHUSETTS02115
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