MATHEMATICS

I. R. SHAFAREVICH

THE EMBEDDING PROBLEM FOR SPLIT EXTENSIONS

(Presented by Academician I. M. Vinogradov on 18 II 1958)

The embedding problem assumes as given a normal extension \(k/\Omega\) with Galois group \(F\), a group \(G\), and an epimorphism \(\varphi : G \to F\). It is required to find conditions under which there exists a normal extension \(K/\Omega\) with Galois group \(G\) such that \(K \supset k\) and the epimorphism \(\varphi\) coincides with the natural homomorphism of the Galois group of the field onto the Galois group of the subfield.

The group \(G\) is called a split extension of its image \(F\) under the homomorphism \(\varphi\) if it contains a subgroup which is mapped isomorphically onto \(F\) under \(\varphi\). There may be several such subgroups. In what follows we shall consider one of them to be chosen and shall denote it by \(F\). If the kernel of \(\varphi\) is \(N\), then \(G = F \cdot N\). We shall say that \(G\) is a split extension of the group \(F\) with kernel \(N\).

The purpose of the present note is to announce the following result:

Theorem 1. The embedding problem is solvable for any field of algebraic numbers \(k\), if \(G\) is a split extension with nilpotent kernel.

This theorem contains, as special cases, a number of results obtained earlier in the embedding problem and in the problem of constructing fields with a prescribed Galois group.

If \(F = 1\), then Theorem 1 proves the existence of an extension with arbitrary nilpotent Galois group \(N\), which had been proved, when the order of \(N\) is odd, by Scholz \((^{1})\) and Reichardt \((^{2})\), and for an arbitrary group \(N\) by the author \((^{3})\). In the case where the kernel \(N\) is abelian, Theorem 1 was proved by Scholz \((^{4})\) and, by a more direct method, by Delone and Faddeev \((^{5})\). In the cases where the group \(N\) is a \(p\)-group of class \(\le p\) or the orders of the groups \(G\) and \(N\) are relatively prime, Theorem 1 was proved by the author \((^{6})\). Since every solvable group \(\mathfrak G\) is a quotient group of a group \(G\) obtained by a chain of split extensions with nilpotent kernels \((^{7})\), Theorem 1 implies the existence of a field of algebraic numbers with an arbitrary solvable Galois group. This fact was earlier proved by the author \((^{8})\) on the basis of the solution of a certain more artificial embedding problem.

The proof of Theorem 1 is based on considerations close to those used in the author’s papers \((^{3,6,8})\). The main difference is that the notion of a Scholz field used in those papers is now replaced by the notion of a relatively Scholz field. It is obvious that one may restrict oneself to the case where \(N\) is a group of order \(l^\alpha\), where \(l\) is a prime number. Let us first consider the case where \(G\) is also an \(l\)-group. The subfield of the field \(K\) belonging to the subgroup \(F\) will be denoted by \(L\).

We shall call the field \(K\) relatively Scholz (relative to \(k\)) if the following conditions are satisfied for it:

1. From each prime divisor of the discriminant of the field \(k/\Omega\) there splits off in \(L\) a prime factor of the first order in the first degree.

1. The prime divisors of the discriminant \(K/k\) decompose in \(K\) into prime divisors of degree 1, and in \(k/\Omega\) are not critical and have order 1.

2. The absolute norms of the prime divisors of the discriminant \(K/k\) satisfy the conditions

\[ \mathfrak N(\mathfrak p) \equiv 1 \pmod {l^h}, \]

where \(h\) is sufficiently large.

1. The prime divisors \(l\) split completely in \(K/k\).

2. The real infinitely remote divisors of \(k\) remain real in \(K\).

The connection of this notion with the embedding problem is based on the following theorem. Let \(Z\) be a normal divisor of order \(l\) in \(N\). Denote by \(\overline N\), \(\overline G\), and \(\overline K\) the groups \(N/Z\), \(G/Z\), and the field having Galois group \(\overline G\). The homomorphism \(\varphi: G \to \overline G\) maps \(F\) isomorphically. Identifying \(\varphi F\) with \(F\), we may write that \(\overline G=\overline N\cdot F\). The subfield of \(\overline K\) belonging to \(\overline N\) will be denoted by \(k\).

Theorem 2. The embedding problem determined by the field \(\overline K\) and the homomorphism \(\varphi: \overline G \to G\) is solvable if the field \(\overline K\) is relatively-Scholz (relative to \(k\)).

Let now the order of \(G\) be arbitrary. Denote by \(H\) the Sylow \(l\)-subgroup of \(G\). Let \(A\) be the minimal Abelian normal divisor of \(G\) lying in \(N\). Put \(G/A=\overline G\), \(H/A=\overline H\), \(N/A=\overline N\), and denote by \(\varphi\) the homomorphism of \(G\) onto \(\overline G\). The field with Galois group \(\overline G\) over the field \(\varkappa\) will be denoted by \(\overline K/\varkappa\), and its subfields belonging to \(\overline H\) and \(\overline N\) by \(\Omega\) and \(k\). Applying the reduction theorem of Faddeev \((^9)\) and Kochendörffer \((^{10})\), one can obtain from Theorem 2 the following result.

Theorem 3. The embedding problem determined by the field \(\overline K/\varkappa\) and the homomorphism \(\varphi: G \to \overline G\) is solvable if the field \(\overline K/\Omega\) is relatively-Scholz (relative to \(k\)).

The field \(K\) which is a solution of the embedding problem formulated in Theorem 3 will not, in general, be relatively-Scholz. Conditions for it to be possible to choose it to be relatively-Scholz can be found analogously to the way this was done in the works \((^{3,6,8})\). They consist in the equality to one of certain invariants \(\chi_i(\overline K)\) of the field \(\overline K\), whose values are roots of \(l\)-th degree of 1.

Suppose that the group \(N\) has \(d\) generators \(s_1,\ldots,s_d\), and that the group \(F\) has order \(m\). Consider the reduced free group of class \(c\), \(\mathfrak M_d^c\), with \(md\) generators \(\sigma_{f,i}\), \(i=1,\ldots,d\), \(f\in F\), whose factor group is \(N\). Defining the permutation \(f\) with \(\sigma_{f,i}\) by the rules

\[ f_1^{-1}\sigma_{f,i} f_1=\sigma_{ff_1,i}, \]

we obtain the group \(F\cdot\mathfrak M_d^c\), which we shall denote by \(\mathfrak G_d^c\). The mapping

\[ f\to f,\qquad \sigma_{f,i}\to s_i^f \]

defines a homomorphism of \(\mathfrak G_d^c\) onto \(G\), mapping \(F\) isomorphically onto \(F\). It follows from this that the solvability of the embedding problem determined by the field \(k\) and the homomorphism \(\mathfrak G_d^c\to F\) implies the solvability of the original embedding problem.

Suppose that for every \(d\) the existence has been proved of a field \(K_d^{c-1}\supset k\) having Galois group \(\mathfrak G_d^{c-1}\). For the possibility of embedding this field into the field \(K_d^c\), it is necessary that the invariants \(\chi^i(K_d^{c-1})\), mentioned above, be equal to 1. Applying the apparatus developed in the works \((^{3,6,8})\), one can obtain the following assertion.

Theorem 4. For every natural number \(\delta\) there exists a \(d(\delta)\) such that in any field \(\overline{K_{\delta}^{\,c-1}}\) containing \(k\) and relatively Scholzian, there exists a subfield \(\overline{K_{\delta}^{\,c-1}}\), also containing \(k\), whose invariants \(\chi_i\bigl(\overline{K_{\delta}^{\,c-1}}\bigr)\) are equal to \(1\).

Such a field \(\overline{K_{\delta}^{\,c-1}}\) can therefore be embedded in a relatively Scholzian (over \(k\)) field \(K_{\delta}^{c}\). The successive application of this process gives the proof of Theorem 1.

CITED LITERATURE

\(^{1}\) A. Scholz, Math. Zs., 42, 161 (1936).
\(^{2}\) H. Reichardt, J. Reine u. Angew. Math., 177, 1 (1937).
\(^{3}\) I. R. Shafarevich, Izv. AN SSSR, ser. matem., 18, 216 (1954).
\(^{4}\) A. Scholz, Math. Zs., 30, 332 (1929).
\(^{5}\) B. N. Delaunay and D. K. Faddeev, Matem. sborn., 15 (57), 243 (1943).
\(^{6}\) I. R. Shafarevich, Izv. AN SSSR, ser. matem., 18, 389 (1954).
\(^{7}\) O. Ore, Duke Math. J., 5, 431 (1939).
\(^{8}\) I. R. Shafarevich, Izv. AN SSSR, ser. matem., 18, 525 (1954).
\(^{9}\) D. K. Faddeev, DAN, 92, 703 (1953).
\(^{10}\) R. Kochendörffer, Math. Nachr., 10, 75 (1953).