S. P. Demushkin

THE EMBEDDING PROBLEM FOR FIELDS OF ALGEBRAIC NUMBERS

(Presented by Academician I. M. Vinogradov on 25 XII 1961)

The embedding problem \((k/\Omega, G, \varphi)\) is considered in the formulation and under the assumptions of the paper \((^1)\); in particular, it is assumed that \(G\) is a \(p\)-group. It is also assumed that \(k\) is a field of algebraic numbers of finite degree.

As is known, a solution of the embedding problem can be constructed by successive \(p\)-steps, which correspond to choosing invariant subgroups of index \(p, p^2\), and so on in the kernel. These successive solutions must be chosen so that the compatibility condition is preserved at each stage. Suppose that \(r-1\) steps can be carried out in the embedding problem in such a way that the compatibility condition is preserved. Denote the field obtained in this way by \(k^{(r-1)}\). Let us determine the conditions under which it is possible to carry out \(r\) steps in the embedding problem while preserving the compatibility condition.

For this purpose consider the embedding problem
\[ \overline{P}_{r-1}\bigl(k^{(r-1)}/\Omega,\, G/A_r,\, \overline{\varphi}_{r-1}\bigr), \]
where \(A_r\) is the invariant subgroup corresponding to the \(r\)-th step. The embedding problem \(\overline{P}_{r-1}\) is solvable, since for the problem
\[ P_{r-1}=\bigl(k^{(r-1)}/\Omega,\, G,\, \varphi_{r-1}\bigr) \]
the compatibility condition is satisfied, and the problem \(\overline{P}_{r-1}\) accompanies it. Let \(k'_r\) be a solution of the problem \(\overline{P}_{r-1}\). For the problem \((k'_r/\Omega, G, \varphi_r)\), compatibility is determined by the algebras \(C_{\chi_i^{(r)}}\) defined by the characters \(\chi_i^{(r)}\) of the group \(A_r\). These algebras, by virtue of the choice of \(k^{(r-1)}\), will be cyclic. Let \(\mathfrak p\) be a prime divisor of the field \(\Omega\) and \(\mathfrak P_{ij}\) its divisors in the field \(k_{\chi_i^{(r)}}\). Take an arbitrary set of integers
\[ c'=\{c_{ij}^{\prime(\mathfrak p)}\} \]
and form the function
\[ J(c')=\prod_{\mathfrak p,i,j}\mu_{\mathfrak P_{ij}}\bigl(C_{\chi_i^{(r)}}\bigr)^{c_{ij}^{\prime(\mathfrak p)}}. \]
Here \(\mu_{\mathfrak P_{ij}}\bigl(C_{\chi_i^{(r)}}\bigr)\) is the invariant of the algebra \(C_{\chi_i^{(r)}}\) at the point \(\mathfrak P_{ij}\) in multiplicative form. This product is meaningful, since only for a finite number of divisors \(\mathfrak P_{ij}\) is the invariant
\[ \mu_{\mathfrak P_{ij}}\bigl(C_{\chi_i^{(r)}}\bigr)\ne 1. \]
The function \(J(c')\) is some \(p\)-th root of unity; generally speaking, it depends on the choice of the fields \(k^{(r-1)}\) and \(k'_r\). We are interested in those
\[ c=\{c_{ij}^{(\mathfrak p)}\} \]
for which \(J(c)\) does not depend on the choice of the fields \(k^{(r-1)}\) and \(k'_r\). Clearly, if \(c\) has this property, then so does
\[ c+pc_1,\qquad pc_1=\{pc_{ij}^{(\mathfrak p)}\}, \]
since always \(J(c'+pc'_1)=J(c')\). Denote by \(H\) the group \((c)/(pc)\), where \((c)\) is the group generated by the elements \(c\).

Theorem. The group \(H\) is finite. In order that there exist a solution of the embedding problem \((k/\Omega, G/A_r, \varphi_r)\), a field \(k^{(r)}\) such that for the embedding problem
\[ P_r=\bigl(k^{(r)}/\Omega,\, G,\, \varphi_r\bigr) \]
the compatibility condition is satisfied, it is necessary and sufficient that all \(J(c)\), \(c\in H\), be equal to unity.

Proof. If the field \(k^{(r)}\) is such that for the embedding problem \(P_r\) the compatibility condition is satisfied, then, taking as \(k'_r\) the field \(k^{(r)}\), we shall have
\[ C_{\chi_i^{(r)}}\sim 1 \]
for all \(\chi_i^{(r)}\). Therefore \(J(c)=1\) for every \(c\in H\). Thus the necessity of the condition of the theorem is proved.

We now show that there are only finitely many elements \(c\). Indeed, fix the field \(k^{(r-1)}\) and vary the field \(k'_r\). This means that we consider the first step for the embedding problem \(P_{r-1}\). As shown in \((^2)\), to each element \(c\in H\) one can put in correspondence a number

\(a_c\) from \(\Omega\) such that \(\Omega\bigl(\sqrt[p]{a_c}\bigr)\subset k^{(r-1)}\). Since all the algebras under consideration will in our case be defined over subfields of the field \(k\), in fact we shall have \(\Omega\bigl(\sqrt[p]{a_c}\bigr)\subset k\). This number is determined up to \(p\)-th powers. But, up to \(p\)-th powers, there are only finitely many of the numbers \(a_c\); therefore there will also be only finitely many elements \(c\).

Let now all \(J(c)\), \(c\in H\), be equal to 1. We shall prove by induction that then there exists a field \(k^{(r)}\) such that, for the problem \(P_r\), the compatibility condition is satisfied. For \(r=1\) the proof of this assertion was given in \((^2)\). Suppose that the assertion is true for \(r-1\). Fix in \(k^{(r-1)}\) a subfield \(k^{(1)}\), and within these limits we shall vary the fields \(k^{(r-1)}\) and \(k'_r\). By the choice of \(k^{(1)}\), for the embedding problem \(P_1\) the compatibility condition is satisfied. For such a problem, the possibility of finding a field \(k^{(r)}\) compatible with \(P_r\) means the possibility of carrying out \(r-1\) steps while preserving the compatibility condition. A necessary and sufficient condition for this is the reduction to 1 of all \(J(c)\) which do not depend on \(k^{(r-1)}\) and \(k'_r\) (\(k^{(r-1)}\) must, of course, contain \(k^{(1)}\)). To reduce such \(J(c)\) to 1 we have only one possibility: to vary the field \(k^{(1)}\). Let us see how \(J(c)\) depends on the field

\[ k^{(1)}=k\bigl(\sqrt[n]{\mu_1 m_1}\bigr),\quad m_1\in\Omega \]

(on \(m_1\) there are also imposed certain conditions, invariant, however, with respect to multiplication). To this end, decompose the algebra \(C_{x_i^{(r)}}\) into a product of two algebras:

\[ C_{x_i^{(r)}}\sim \overline{C}_{x_i^{(r)}}\otimes \widetilde{C}_{x_i^{(r)}}, \]

where \(\overline{C}_{x_i^{(r)}}\) is some fixed algebra of the type \(C_{x_i^{(r)}}\), and \(\widetilde{C}_{x_i^{(r)}}\) is the algebra computed for the split group extension. The function \(J(c)\) will therefore be represented in the form of a product

\[ J(c)=\overline{J}(c)\,\widetilde{J}(c). \]

Since \(J(c)\) does not depend on the choice of the fields \(k^{(r-1)}\) (\(k^{(r-1)}\supset k^{(1)}\)) and \(k'_r\), it follows that \(\widetilde{J}(c)\) does not depend on such a choice. Applying to the algebra \(\widetilde{C}_{x_i^{(r)}}\) the generalization of Theorem 2 from \((^1)\) on multiplication of algebras, we obtain that \(\widetilde{J}(c)\) depends on \(m_1\) multiplicatively. Let us write this dependence explicitly:

\[ J(c)=\overline{J}(c)\,\widetilde{J}(c,m_1). \]

We need to achieve that \(J(c)=1\). There are only finitely many such \(\overline{J}(c)\). Let these be \(J(c_1), J(c_2),\ldots,J(c_t)\). It is necessary, therefore, that the equations

\[ \overline{J}(c_i)\,\widetilde{J}(c_i,m_1)=1 \]

be solvable, or

\[ \widetilde{J}(c_i,m_1)=\overline{J}(c_i)^{-1}. \]

We shall show that, under our assumptions, this system of equations is solvable. Indeed, the solvability condition for such a system of equations is the following property: if

\[ \prod_i \widetilde{J}(c_i,m_i)^{a_i}=1 \]

for all \(m_i\), for some set of integers \(a_i\), then it must be that

\[ \prod_i \overline{J}(c_i)^{-a_i}=1, \]

or

\[ \prod_i \overline{J}(c_i)^{a_i}=1. \]

Suppose

\[ \prod_i \widetilde{J}(c_i,m_1)^{a_i}=1. \]

This means that

\[ \widetilde{J}(c)=\prod_i \widetilde{J}(c_i,m_1)^{a_i}=1, \]

where

\[ c\approx \prod_i c_i^{a_i} \]

(equality up to \(p\)-th powers), does not depend on the choice of the fields \(k^{(r-1)}\) and \(k'_r\). Since

\[ J(c)=\overline{J}(c)\,\widetilde{J}(c), \]

it follows that \(J(c)\) also does not depend on the choice of \(k^{(r-1)}\) and \(k'_r\). The latter means that, by the condition of the theorem, it must be that \(J(c)=1\); then

\[ J(c)=\overline{J}(c)\,\widetilde{J}(c)=1. \]

The theorem is proved.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
25 XII 1961

REFERENCES

1. S. P. Demushkin, I. R. Shafarevich, Izv. AN SSSR, ser. matem., 23, 823 (1959).
2. S. P. Demushkin, I. R. Shafarevich, Izv. AN SSSR, ser. matem., 26, No. 6 (1962).