UDC 519.49

MATHEMATICS

Z. I. BOREVICH

ON GROUPS OF PRINCIPAL UNITS OF \(p\)-EXTENSIONS OF A LOCAL FIELD

(Presented by Academician Yu. V. Linnik on 14 V 1966)

Let \(k\) be a finite extension of the field of \(p\)-adic numbers of degree \(n\) (a local field), and let \(K/k\) be a normal \(p\)-extension with Galois \(p\)-group \(G\) of order \(m\). The group of principal units \(E\) of the field \(K\) is a multiplicatively written module over the group ring \(O=Z[G]\) of the group \(G\) with coefficients in the ring of \(p\)-adic integers \(Z\). It is of known interest to determine the structure of the \(O\)-module \(E\).

We shall call two finitely generated \(O\)-modules similar if they differ from each other by a direct \(O\)-summand that is a free \(O\)-module. Since the Krull–Schmidt theorem is valid for finitely generated \(O\)-modules (see \((^{1})\), § 8), in studying the \(O\)-module \(E\) we may consider any \(O\)-module similar to it.

A prime element \(\Pi\) of the field \(K\) can be chosen so that \(\sigma(\Pi)/\Pi=\theta_\sigma\in E\) for all \(\sigma\in G\). We pass in the group \(\{\Pi\}\times E\), invariant with respect to the operators from \(G\), to additive notation and embed it in an \(O\)-module \(A\), allowing as coefficients at \(\Pi\) arbitrary integral \(p\)-adic numbers (and not only rational integers). The structure of the \(O\)-module \(A\) may, in a certain sense, be considered known; see \((^{2,3})\).

By \(I\) we denote the submodule of the ring \(O\) generated by the elements \(\sigma-1\) \((\sigma\in G)\).

Theorem 1. The \(O\)-module \(E\) is similar to a certain \(Z\)-splitting extension of the \(O\)-module \(A\) by means of \(I\).

Indeed, for the direct sum \(X=E\oplus Ox\) define \(O\)-homomorphisms \(i:I\to X\) and \(j:X\to A\), putting

\[ i(\sigma-1)=-\theta_\sigma+(\sigma-1)x;\qquad j(\varepsilon)=\varepsilon\;(\varepsilon\in E),\qquad j(x)=\Pi. \]

Then we shall have an exact sequence

\[ 0\to I\overset{i}{\to}X\overset{j}{\to}A\to 0. \tag{1} \]

Let \(A\) and \(C\) be arbitrary \(O\)-modules and let \(0\to C\overset{i}{\to}X\overset{j}{\to}A\to 0\) be some \(Z\)-splitting extension. Define a \(Z\)-homomorphism \(l:A\to X\) so that \(jl=1\). For \(\sigma\in G\) and \(a\in A\) put \(f(\sigma)(a)=i^{-1}(\sigma l(\sigma^{-1}a)-l(a))\). This defines a 1-cocycle \(f\) on \(G\) with values in the group \(\operatorname{Hom}_Z(A,C)\). It is easily verified that the correspondence \(X\to f\) defines a natural isomorphism of the group \(\operatorname{Ext}^{*}(A,C)\) of classes of equivalent \(Z\)-splitting extensions of the \(O\)-module \(A\) by means of \(C\) onto the group \(H^{1}(G,\operatorname{Hom}_Z(A,C))\).

In view of Theorem 1, to determine the \(O\)-module \(E\) one must find, in the group \(\operatorname{Ext}^{*}(A,I)\), which we identify with \(H^{1}(G,\operatorname{Hom}_Z(A,I))\), the element corresponding to the extension (1).

Let \(Q\) be the periodic part of the \(O\)-module \(A\); \(A_0=A^G\) the subgroup of \(G\)-invariant elements in \(A\); \(\Delta\) the group of norms \(N(A)\) of elements of \(A\), and \(D\) the set of those elements of \(A\) some multiple of which falls into \(A_0\). For a finite \(p\)-group \(\mathfrak A\) we denote the group \(\operatorname{Hom}(\mathfrak A,R^{+}/Z)\), where \(R^{+}\) is the additive group of all \(p\)-adic numbers, by \(\operatorname{Char}\mathfrak A\), and its

elements characters of the group \(\mathfrak A\). Let \(v\) be the valuation of the field \(K\), considered as an element of the group \(\operatorname{Hom}(A,Z)\).

Theorem 2. There is a natural isomorphism
\[ \operatorname{Ext}^*(A,I)\approx \operatorname{Char}(D/(\Delta+Q)). \]
Under this isomorphism the extension (1) corresponds to the character \(\chi\) for which
\[ \chi(u \bmod (\Delta+Q))=\frac{v(u)}{m}\bmod Z\qquad (u\in D). \]

Denote by \(f\) the degree of inertia of the extension \(K/k\).

Theorem 3. There is an exact sequence
\[ 0\to \operatorname{Char} H^1(G,Q)\xrightarrow{\lambda}\operatorname{Ext}^*(A,I)\xrightarrow{\mu}\operatorname{Char}G \tag{2} \]
with natural homomorphisms \(\lambda\) and \(\mu\). If an element \(c\in \operatorname{Ext}^*(A,I)\) corresponds to the extension (1), then the kernel of the character \(\mu(c)\in \operatorname{Char}G\) coincides with the inertia subgroup \(G_0\) of the extension \(K/k\), and
\[ \mu(c)(\sigma)=\frac{1}{f}\bmod Z \]
for an automorphism \(\sigma\in G\) inducing on the inertia subfield the Frobenius automorphism. Furthermore, \(\mu(c)\) coincides with the composite homomorphism of the canonical sequence
\[ G\to H^0(G,A)\xrightarrow{\delta}H^1(G,I)\to R^+/Z. \]

Suppose that \(K/k\) is a totally ramified extension. Since in this case \(\mu(c)=0\), there exists a uniquely determined element
\[ \varphi\in \operatorname{Char}H^1(G,Q) \]
such that \(\lambda(\varphi)=c\). The character \(\varphi\) is determined in terms of the extension \(K/k\) as follows. Let
\[ [g]\in H^1(G,Q) \]
be a cohomology class determined by a cocycle \(g\in Z^1(G,Q)\). For some element \(u_g\in D\) we have
\[ g(\sigma)=(\sigma-1)u_g. \]
Put
\[ \varphi([g])=\frac{v(u_g)}{m}\bmod Z. \]

Theorem 4. If \(K/k\) is a totally ramified \(p\)-extension, then for the just-defined character \(\varphi\) from the group \(\operatorname{Char}H^1(G,Q)\) we have \(\lambda(\varphi)=c\).

The extension (1) will be \(O\)-split if and only if \(c=0\). This gives us the following result.

Theorem 5. If \(K/k\) is a totally ramified \(p\)-extension and if the maximal subfield of \(K\) radical over \(k\) is generated by adjoining roots of principal units of the field \(k\), then the group \(E\), as an \(O\)-module, is similar to the direct sum \(I\oplus A\).

Let us now consider the case of a regular local field \(k\) (not containing a primitive root of degree \(p\) from \(1\)). For regular \(k\), the exact sequence (2) reduces to the isomorphism
\[ \operatorname{Ext}^*(A,I)=\operatorname{Ext}(A,I)\approx \operatorname{Char}G. \]

Theorem 6. If \(k\) is a regular local field, then the \(O\)-module \(E\) is similar to the tensor product \(I\otimes Y\) (over \(Z\)), where the \(O\)-module \(Y\) is determined as the extension
\[ 0\to Z\to Y\to I\to 0, \]
corresponding to a character
\[ \chi\in \operatorname{Char}G\approx \operatorname{Ext}(I,Z), \]
whose kernel coincides with the inertia subgroup \(G_0\) of the extension \(K/k\), and for which
\[ \chi(\sigma)=\frac{1}{f}\bmod Z \]
for an automorphism \(\sigma\in G\) inducing on the inertia subfield the Frobenius automorphism.

In the paper [4], Theorem 6 was established (by another method) for the case when the minimal number of generators of the group \(G\) does not exceed \(n\). Keeping the regularity condition on \(k\), suppose that the minimal number of generators of the group \(G\) is equal to \(d+1\), and that its generators \(\sigma_0,\sigma_1,\ldots,\sigma_d\) are chosen so that
\[ G/[G,G]=\{\overline{\sigma}_0\}\times\{\overline{\sigma}_1\}\times\cdots\times\{\overline{\sigma}_d\} \]
(here \(\overline{\sigma}_i\) denotes the coset modulo the commutator subgroup \([G,G]\) with representative \(\sigma_i\)). Let

\(a_i\) is the order of the element \(\bar{\sigma}_i\) \((0\leq i\leq d)\). Consider the free \(O\)-module \(W\) of rank \(d+1\) with free \(O\)-generators \(x_0,x_1,\ldots,x_d\). The mapping \(x_i\mapsto \sigma_i-1\) defines an operator epimorphism \(W\to I\). Denote its kernel by \(\Omega\). The module \(\Omega\) is indecomposable\({}^{4}\) into a direct sum of \(O\)-modules. The elements

\[ l_i=\sum_{\tau\in G}\tau(x_i)\quad (0\leq i\leq d) \]

form a basis of the \(Z\)-lattice \(\Omega^G=W_G\).

Denote by \(\Theta\) the kernel of the mapping

\[ N:\ x\mapsto N(x)=\sum_{\tau\in G}\tau(x),\quad x\in\Omega. \]

The \(O\)-module \(\Theta\) is also indecomposable. Since the elements \(a_i l_i\) \((0\leq i\leq d)\) form a basis of the \(Z\)-lattice \(N(\Omega)\), we have \(a_i l_i=N(v_i)\) for some \(v_i\in\Omega\). It is obvious that the module \(\Omega\) is generated by the submodule \(\Theta\) and the elements \(v_i\), i.e.
\[ \Omega=\{\Theta,v_0,v_1,\ldots,v_d\}. \]

We may assume that \(\sigma_0\) induces the Frobenius automorphism on the inertia subfield. Let the rational integers \(r_i\) \((1\leq i\leq d)\) be chosen so that \(\sigma_i\sigma_0^{-r_i}\in G_0\). Then
\[ G_0=\{\sigma_0^f,\sigma_1\sigma_0^{-r_1},\ldots,\sigma_d\sigma_0^{-r_d}\}. \]
Put
\[ c_0=a_0/f,\quad c_i=a_i r_i/f\quad (1\leq i\leq d). \]
In the direct sum \(\Omega\oplus Ow\), consider the submodule
\[ M=\{\Theta,Iw,v_0+c_0w,v_1+c_1w,\ldots,v_d+c_dw\}. \]

Theorem 7. For regular \(k\), the group of principal units \(E\) of the field \(K\), as an \(O\)-module, is similar to the \(O\)-module \(M\) constructed above. If \(f>1\) and \(c_i\equiv0\pmod p\) for all \(i=0,1,\ldots,d\), then the \(O\)-module \(M\) is indecomposable. If, however, at least one of the numbers \(c_i\) is not divisible by \(p\), then \(M\) decomposes into a direct sum of an indecomposable \(O\)-module and a free \(O\)-module with one generator.

Thus, if \(f>1\), the group \(E\) decomposes into a direct sum of a free \(O\)-module and a certain indecomposable \(O\)-module \(E_0\). The \(Z\)-rank of the module \(E_0\) is determined by the following theorem.

Theorem 8. If \(K/k\) is not a totally ramified extension of a regular local field \(k\), then the group of principal units \(E\) of the field \(K\), as an \(O\)-module, is similar to an indecomposable \(O\)-module \(E_0\), which is a \(Z\)-lattice of rank \((d+1)m\), if the inertia subfield inside \(K\) is embeddable in a cyclic ramified extension over \(k\), and of rank \(dm\) in the opposite case.

Leningrad State University
named after A. A. Zhdanov

Received
28 IV 1966

CITED LITERATURE

\({}^{1}\) Z. I. Borevich, D. K. Faddeev, Vestn. Leningradsk. Univ., No. 7, 72 (1959).
\({}^{2}\) Z. I. Borevich, Izv. AN SSSR, Ser. Matem., 16, No. 5, 427 (1952).
\({}^{3}\) Z. I. Borevich, Uch. Zap. Kabardino-Balkarsk. Gos. Univ., Ser. Fiz.-Matem., vol. 24, 53 (1965).
\({}^{4}\) Z. I. Borevich, Tr. Matem. Inst. im. V. A. Steklova AN SSSR, 80, 30 (1965).