proceedings of the
american mathematical society
Volume 118, Number 3, July 1993
A NOTE ON HILBERT'S THEOREM 90
BAO-PINGJIA AND LARRYSANTONI
(Communicated by Maurice Auslander)
Abstract. In this paper we extend "up to powers" Hubert's Theorem 90 to
arbitrary finite Galois extensions. In the case of algebraic number fields with
class number equal to 1, we completely determine the kernel and image of the
norm map.
Let L/K be a finite Galois extension with Galois group G. Hilbert's Theorem 90 gives us a characterization of the kernel of the norm map in the case
where L is a cyclic extension, i.e., if G is a cyclic group. In this paper, we
investigate the module structure of the multiplicative group of field extensions
as ZCr-modules and their associated norm functions. We obtain some partial
results about the image and the kernel of these norm functions. For example, by
using a theorem from [J] (see (1.6) in §1 below), we find the kernel and image
of 1q<8>zN
, where N:L*-»7C* is the norm map 1q®zN: Q®zL* -* Q®z^*
(where L* denotes the nonzero elements of L). We will omit the subscript
Z in the tensor product notation as is conventional. We have completely determined the kernel and image of the norm map in the situation of algebraic
number fields with class number equal to 1. We also obtain a partial generalization of Hilbert's Theorem 90 to arbitrary finite Galois field extension, not
necessarily cyclic.
1. Hilbert's
Theorem
90
Let L/K be a finite Galois extension with Galois group G, and let ZC7 be
the group ring. If a £ L* and g £ G, we write ag instead of g(a). Since a"
is the rath power of a as usual, in this way L* becomes a right ZG-module in
the obvious way. For example, if r = 3g + 5 G ZC7, then of = (a$)g(as). For
any co £ L*, the norm of co is defined to be N(co) = UL/K(co) = Ili^lc? s G}.
If we let c = Y,{g\8 e G} , then N(w) = coc.
Before going further we first quote some results from [J].
1.1. Theorem. Let L/K be a finite Galois extension with Galois group G, and
assume that K is not an algebraic extension of a finite field. Then Q ® L* is a
free QG-moduleof rank card(A^).
Received by the editors November 13, 1991.
1991 Mathematics Subject Classification. Primary 12F10; Secondary 12F05, 11R27.
©1993 American Mathematical Society
0002-9939/93 $1.00+ $.25 per page
739
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740
B.-P. JIA AND LARRY SANTONI
1.2. Corollary. L* is a ZG-module, and there is a free ZG-module F con-
tained in L* such that:
(1.2.1) L*/F is a torsion abelian group;
(1.2.2) a £ Ff\K <* a = N(B) for some 0 £ F; i.e., image(N|f) = FnK
and
kcr(N\F) = UF(l-g)\g£G};
(1.2.3) For each a £ L*, there exists an integer m such that am £ K •*=>
am =
NO?)for some B £ L*.
Proof. If K is algebraic over a finite field, then L* is torsion and we let F = 1.
If K is not algebraic over any finite field, then, by (1.1), Q®L* is a free QGmodule of rank card(AT). One checks that if b = £ r,<8>}>,
is a basis element for
Q®L*, then it may be rewritten as b = ^®y for a suitable choice of d £ Q and
y £ L*. Then let F be the free ZG-module generated by the y 's and (1.2.1)
follows. For (1.2.2), let c = £{g|g G G}. If a = N(/J) for some 0 £ F, then
a= Bc £Ff)K.
If 1 ^a£FnK,
then ac = a" where ra = [L : K]. Since F
is free, we may assume that a = ni^8,y\S
& G} where y is a basis element
of F and ag,,, is an integer for each g and each y . Then for each fixed y ,
nag,y = 12iah,y\h s C7}for each g, and we let ay —agyy = ahy for all g and
« . Let B = n'{y^} . Then a = N(/i) = j?c. The theorem follows.
1.3. Theorem. Let L/K be a finite Galois extension with Galois group G,
and assume that K is not an algebraic extension of a finite field (otherwise
Q <8>
L* = 0). Let H be any subgroup of G, and let LH = {a £ L\ah —a for
all h£H}
be the fixedfield of H and cH = £{«|« £ 77}. Then Q®L*H*i
((QG^W^Ch
= (QC7c«)card^).
Proof. Since Q®L* = (Qc7)card(/i:),with some abuse of notation, we may iden-
tify 1 ® a with its image in QC7. For any a £ L*H, (1 ® a)h = (1 ® a*) =
(1 ® a).
Without loss of generality, we may assume
1 ® a £ QG.
Then
1 g) a = zZiagg\h 6 G and ag £ Q}. Write G = \J{giH\l < i < r}, where
r = [G: H]. Then 1 ® a = ERv!c?<«l« e 77, 1 < / < r and a/A€ Q} . Since
(1 g) a)h = (1 ® a), for each i, there exists a, G Q such that a, = a,/, for any
ra G 77. Therefore 1 ® a G QC7cwand Q ® L^ c (QGc//)03^^). Conversely,
we choose F = (ZG)caTd^ as in (1.2) such that Q<g>L*= (QG)'"*™ = Q<8>F.
Then FCQ®f
= Q®L* = (QC7)card^).If A GQGc// C QG, then y?= i ®a
for some a£F.
Now ph — p implies that ara = a since T7 is free. Therefore
a£L*H and (QGc//)card<*:)
cQxLJ,.The
theorem follows.
1.4. Corollary. Let K/k be a finite separable extension. Suppose k is not
algebraic over a finite field. Let L be the normal closure of K, and let G -
Ga\(L/k). Then K = Lh is the fixed field of 77 for some subgroup H of G
and Q ® K* * ((QG)™*™)^ = (QGch)™*™, where cH= ^{h\h £ 77}.
If G is generated by elements gx, ... , gm , then, in the group ring ZG, for
any g £ G, 1 - g £ £{ZG(1 - gt)\ 1 < i < m}—the left ideal of ZG generated
by the gi's. The reason is that any element of G is a product of powers of the
gi's. For example,
i-gf = (i + gi + •■■
+ gf-l)(i -gd e z2iZG(l -S')!1 <*<"*}■
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A NOTE ON HILBERT'STHEOREM90
741
Therefore
1 - g?g?gV = 1 - tf + g? - *,'*? + gt'gl ~ g?g2g?
= (l-g?)
+ g?(l-
6 J]{ZG(l-ft)|l
c?22)+ *,'#(!
- c?D
<I<lfl}.
The following corollary is an easy consequence of (1.2.2).
1.5. Corollary. If G is generated by elements gi, ... , gm, then, in the group
ring ZG, Y,{ZG(l - g,)|l < i < m} is a left ideal of ZG generated by the
elements gt 's. For each g£G, 1 - g £ £{ZG(1 - gt)\\ < i < m}. Let F be
as in (1.2). Then ker(N|f) = £{7^(1 - &)|1 < i < m}.
1.6. Theorem. Let L/K and G be as in (1.1). Suppose G is generated by
elements gi, ... , gm. If n £ L* and N(ra) = 1, then nn = UiP!~g'\ * <i<m}
for some fij £ L* and n = [L : K].
Proof. Suppose 1 = N(w) = nc. Therefore ra = rii"'0-^!1
ra"= Il{'/(1_*)lc?e G} . The result follows from (1.5).
# g e G} and
1.7. Theorem (Hilbert's Satz 90). Let L/K be a cyclic extension, and let n be
a generator of the Galois group of L/K. Let N = IM^ . Then N(h) = 1 for
u £ L if and only if there exists a v £ L such that u —v/vn (sometimes written
as v1^).
1.8. Corollary. Let L/K be a Galois extension with Galois group G. Suppose
G is generated by elements g\,... , gm. Then
(1.8.1) Q<8>Ker(N) S pard^
1 - g,i, l<i<m;
(1.8.2) Q®N(L*)^c(QG)card(L)
wnere j is tne \eft idea\ 0f qq
generated by
= (Qc)card(L),where c = E{c?lc? e G};
(1.8.3) Q 9 (L*/K*) S 7?card^, where R = QG/Qc.
Proof. (1.8.1) followsfrom (1.6), and (1.8.2) is trivial since QGc = Qc. (1.8.3)
follows from the fact that if a £ K, then a" = N(a) with n - [L : K], hence
i<g>a=l<g>a" = l® N(a) and Q ® N(7_*)D Q ® A"*.
2. Some results in algebraic number theory
In this section we discuss some problems in algebraic number theory. A
number of well-known results, which are quoted here, can be found in [K] or
[M].
Let L/K be a Galois extension with Galois group G. It is clear that the
restriction of the norm function N = NL/K to the multiplicative group L* of
nonzero elements of L defines a homomorphism of L* to K*. It is natural
to seek information on the image N(L*) and kernel of N: L* —>K*. This
information is usually not easy to obtain. In the case where the class number of
L is equal to 1, we are able to answer this question. We also have some partial
results in the general case.
2.1. Notation. Let K be a number field, and let A be the integral closure of
Z in K. Then A is a Dedekind domain. Let L/K be a finite Galois extension
with Galois group G. Let B be the integral closure of A in L. Then L is
the quotient field of B . For each nonzero prime ideal p in A , we can choose
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742
B.-P. JIA AND LARRY SANTONI
just one nonzero prime ideal P of B such that P n A = p . Let P be the set
of all such nonzero prime ideals in B, and let Vl/k - {^-adic valuation on
LIT' G P} , and \L/K = {p-adic valuation on K\p = P C\A and P £ P}. Since
each nonzero element of B is contained in only finitely many nonzero prime
ideals of B, the set VL/K aas finite character, i.e., for any nonzero element
a £ L, only finitely many members of VL/K are nonzero at a. We also know
that card(V) = card(K) = card(L) since both V and K are countable. Let
v £ ?l/k , and let V £ Yl/k designate the corresponding extension of v . We
define valuations Vs for g £ G by the formula Vg(a8) = V(a) for all a £ L*.
(Equivalently, Vg(a) = V(ag~l).) Before proceeding to our main results, we
first prove the following easy lemma.
2.2. Lemma. Let K, A, L, B, and G be as in (2.1), and let U be the unit
group of B. Let a, B £B-{0}.
Then Vg(a) = Vg(P) for all g £ G and all
V G Vl/k & a = ufi for some u £ U.
Proof. We let 7 = q77 and J = pB. Then Vg(a) = V*(B) for all g £ G and
all V £ VLiK «• 7 and J have the same prime factorization in B <=>7 = J «■
a = up and P = wa for u, w £ B. But then a = up = uwa,
and we have
uw = 1 and u£ U.
2.3.
The map <j>induced by v = yL/K . For each v £ v, we define a homo-
morphism <f>v:
L* -> ZG by
(2.3.1)
&,(<*)
= £{(r*(«))*l* e (7}.
It is easy to check that </>vis a ZG-module homomorphism. Since VL/K has
finite character, the ZG-module homomorphisms </>vinduces a ZG-module
homomorphism
<j>— @{<f>v\v£ v} from L* to (ZG)card^v^,the direct sum of
card(v) copies of ZG. We want to find the image and the kernel of <j>. Let
c be the sum, in ZG, of the elements of G. Then Zc is a ZG-submodule
of ZG and is also an ideal of ZG. We note that since 0 is a ZG-module
homomorphism
from L* to (ZG)card(v), <f>commutes with c.
2.4. Theorem. Let K, A, L, B, and G be as in (2.1), and let U be the unit
group of B. Let c be as in (2.3). Then
(2.4.1) ker(</3)= U and imaged) is a ZG-submodule of (ZG)caTdW
;
(2.4.2) L* =■U ®<f>(L*)as Z-submodules;
(2.4.3) 4>(N(L*)) = c<f>(L*)C c(ZG)card(v)= (Zc)card<*>.
Proof. For each x £ L*, since L is the quotient field of 77, we can write
x = a/P for some a, P £ B - {0}. Then cj>(x)= 0 <=><f>v(x)- 0 for all
i/ev«
Vg(a) = Vg(p) for all g £G and for all KeV«a
= up for some
u £ U. Since <£ is a ZG-module homomorphism and <p(L*) C (ZG)card(v),
<j>(L*)is a free Z-module. Hence L* = U ® <j>(L*)as Z-modules (not as
ZG-modules).
2.5. Theorem. Let K, A, L, B, c, U, and G be as in (2.4), and let n be
the class number of L. Let S = {ra'|0 < / < oo}, and let Zs - S~'Z
be the
localization of Z at S. For any subgroup 77 of G, let cH = Y,{h\h £ H} £ ZG.
Then
(2.5.1) Zs ® <p(L*) is a direct sum of cyclic right ZsG-modules on generators of
the form ch , where 77 is a subgroup of G;
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A NOTE ON HILBERT'STHEOREM90
(2.5.2) Zs ® f/3(N(L*))-Zs®
rncZs,
743
c(<f>(L*))is a direct sum of cyclic ZsG-modules
where rn —card(77) for the subgroup H of G and
Zs <g>
<t>(U(L*)) c c(ZG)card(v) = (Zc)card(T).
Proof. Since n is the class number of L, P" is a principal ideal of 77 for any
prime ideal P of 77. Let 77 = {g £ G\Pg = P} be the decomposition group
of P, and let V be the 7>-adicvaluation. Let <pvbe defined by (2.3.1). Then
Izs ® 0 is a surjective ZG-module homomorphism from Zs ® L* to a sum of
cyclic modules of the form CuZsG, and the theorem follows from this.
2.6. Corollary. Let K, A, L, B, c, U be as in (2.4). Suppose the class number
of L is equal to 1. Let G be generated by gx, ... , gm- For any P £ P, let
HP = {g£ G\Pg = P}. Then
(2.6.1) <f>(L*)
= ®{cHpZG\P£ P} and L* ^ U ® <p(L*)as ZG-modules;
(2.6.2) N(L*)^A®c<f>(L*),where AC {1,-1};
(2.6.3) c<f>(L*)
= ®{rHp(cZ)\P£P}.
Let M be the subgroup of Q generated by {pd(p)\p is a positive prime number
of Z}, where d(p) = o(HP) is the degree of the prime ideal P with 7>nZ = pZ.
Then
(2.6.4) N(L*) = A®M and M is torsionfree.
Let U(l) = {u £ U\N(u) = 1}. For each PeP,
choose nP £ P such
that P - KpB. Let D be the ZG-submodule of V generated by {naP\a=
P - pgj, p £ ZG}, and let F be the submodule of L* generated by all such
up 's. Then
(2.6.5) F =■(f>(L*)= ${cHpZG\P £ P}, and L* = UF =■U ® F is the direct
product of U and F;
(2.6.6) ker(N) = U(l)D and D is torsionfree.
Proof. Since the class number of L/K is 1, 0 must be surjective and (2.6.1)(2.6.5) follow. For (2.6.6), we need only check that the annihilator of c in
cp(L*) is <t>(D).As in the proof of (1.3), write G = \J{giH\\ < i < r}, where
r = [G : 77]. If 0 = (cff 5>&)c = c„(£«, - »,(1 - gi))c, then £0,- = 0
since c annihilates the 1 - g,■.
Let r and 2.$ denote the number of real and nonreal automorphisms of L.
Then the Dirichlet unit theorem tells us that U = W ® V , where W is a finite
cyclic group consisting of the roots of 1 in L and V is a free abelian group of
rank r + s - 1. Then U is a finitely generated Z-module and every subgroup
of U is finitely generated as an abelian group.
2.7. Lemma. Let U(l) be as in (2.6). Then U(\) s W(\)®V(\),
where
W(\) = (co) is a finite cyclic group generated by some root co of 1 in L and
V(l) = ©{(m/)|1 < i < t} is a free abelian group of rank t < r + s - 1.
2.8. Corollary. Let L and U be as in (2.6). Let U(l) Si (co)® {©{Zw,|l <
i < t}} be as in (2.7), and let D be as in (2.6). Then Ker(N) = U(l)D £
(o)) ® {0{Z«,| I < i <t}} ®D is a direct sum.
Remark. A complete generalization of Hilbert's Theorem 90 can now be obtained in the case where the «, 's can be written in the form «, = Px~8. We
do not know at this point whether this is always possible.
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744
B.-P. JIA AND LARRY SANTONI
The results in this paper suggest that a unique factorization is possible even
though the field is not the quotient field of a unique factorization domain. For
example, (1.2) tells us that for any finite Galois extension L/K with Galois
group G, there is a free ZG-module F contained in L* such that L*/F is a
torsion abelian group. It follows that some power of any nonzero element of L
can be factored with unique exponents in ZG.
References
[H]
[J]
D. Hilbert, Theorie der AlgebraischeZahlkorper, 1897, p. 147.
Bao-Ping Jia, Splitting of rank-one valuations, Comm. Algebra 19 (1991), 777-794.
[K]
Gregory Karpilovsky, Unit groups of classical rings, Oxford Science Publ., Clarendon Press,
[M]
Oxford, 1988.
Daniel A. Marcus, Number fields, Springer-Verlag,New York, 1977.
Department
Alaska 99775
of Mathematical
Sciences, University
of Alaska Fairbanks,
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Fairbanks,