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MATHEMATICS
HELMUT KOCH
THE GALOIS GROUP OF A LOCAL FIELD
(Presented by Academician I. M. Vinogradov on 1 XII 1960)
Let \(k\) be an extension of finite degree \(m\) of the field \(R_p\) of rational \(p\)-adic numbers. The field \(k\) is called regular if it does not contain a \(p\)-th root of unity.
Let \(k\) be a regular field. In the work of I. R. Shafarevich \((^1)\) it is shown that if \(K\) is the union of all normal finite \(p\)-extensions of the field \(k\), then the group \(\Phi\) of the field \(K\) over \(k\) is a free topological \(p\)-group with \(m+1\) generators.
Z. I. Borevich \((^2)\) considered the case where \(K\) is the maximal extension without simple branching of the field \(k\). The result of his work is that \(\Phi\) is a free group with another topology. But in doing so he did not notice that an unramified extension of a regular field is not always regular. For example, let \(k = R_3(\sqrt{3})\); then \(k(\sqrt{-1})\) is an unramified irregular extension of the regular field \(k\). Therefore Borevich’s result will be valid only in the case when \(K\) is a regular field.
Let us now consider the totality of fields \(K\) having the following property: every finite regular extension of the field \(k\) is contained in one of the fields \(K\).
Let \(K_0\) be a finite regular normal extension of the field \(k\) without higher ramification, having a solvable Galois group. \(K_0\) has the representation
\[ K_0 = k\left(\sqrt[e]{\pi}, \zeta\right), \qquad e \equiv 0 \pmod{q^f - 1}, \]
where \(\pi\) is a prime element of the field \(k\); \(\zeta\) is a \(q^f\)-th root of unity; \(q\) is the number of elements of the residue class field of the field \(k\).
Let \(K\) be the union of all finite normal \(p\)-extensions of the field \(K_0\); \(F\) a free group with \(m+2\) generators \(s, t, a_1, \ldots, a_m\); \(W = (t^e, sts^{-1}s^{-q})\) and \(F_0 = (W, s^f, a_1, \ldots, a_m)\) normal divisors of the group \(F\). Then the following holds:
Theorem 1. The group \(\Phi\) of the field \(K\) over \(k\) is isomorphic to the completion \((F/W)^*\) of the group \(F/W\) with respect to the topology defined by the system of neighborhoods of the identity consisting of the set of normal divisors \(N\) of the group \(F/W\) that satisfy the conditions \(N \subseteq F_0/W\) and \([F_0/W : N]\) is a power of \(p\).
Proof. We construct the fields \(K_n\) inductively. Let \(K_{n+1}\) be the union of all normal extensions of degree \(p\) of the field \(K_n\). Then \(K_{n+1}\) will be a normal field over \(k\) and will be the class field of the field \(K_n\) for the group \((K_n^*)^p\) (\(K_n^*\) denotes the multiplicative group of the field \(K_n\)). Let \(\Phi_n\) be the group of the field \(K_n\) over \(k\). \(K\) is the union of all fields \(K_n\).
The group \(\Phi_0\) is generated by two automorphisms:
\[ \sigma=\left(\zeta\to \zeta^q,\ \sqrt[e]{\pi}\to \sqrt[e]{\pi}\right); \tag{1} \]
\[ \tau=\left(\zeta\to \zeta,\ \sqrt[e]{\pi}\to \zeta^{\frac{q^f-1}{e}}\sqrt[e]{\pi}\right). \tag{2} \]
\(\sigma\) and \(\tau\) satisfy the relations \(\sigma^f=1,\ \tau^e=1,\ \sigma\tau\sigma^{-1}=\tau^q\). Our next aim is the computation of the group \(\Phi_1\). Let \(\eta\in\Phi_0\), and let \(\eta_1\) be its extension to the field \(K_1\).
For the norm residue symbol \(\left(\dfrac{K_1/K_0}{\alpha}\right)\) the following equality holds:
\[ \left(\frac{K_1/K_0}{\alpha^\eta}\right) =\eta_1\left(\frac{K_1/K_0}{\alpha}\right)\eta_1^{-1}, \qquad \alpha\in K_0^* . \tag{3} \]
Krassner \(\left({}^{3}\right)\) showed that every \(\alpha\in K_0^*\) has a representation
\[ \alpha=\sqrt[e]{\pi^{\,a}}\zeta^b\prod_{\nu=1}^{m}\varepsilon_{\nu}^{\,a_{\nu,\eta}}, \tag{4} \]
where \(\varepsilon_1,\ldots,\varepsilon_m\) is a system of independent principal units of the field \(K_0\); \(a\) and \(b\) are integral rational numbers, and \(a_{\nu,\eta}\) is an integral \(p\)-adic number.
One can find representatives \(\sigma_1,\tau_1\) of the automorphisms \(\sigma,\tau\) which satisfy the relations
\[ \sigma_1^f=\left(\frac{K_0'/K_0}{\sqrt[e]{\pi^{g}}\varepsilon}\right), \qquad g\not\equiv 0\ (p),\quad \varepsilon\in K_0 \text{ a unit}; \tag{5} \]
\[ \sigma_1\tau_1\sigma_1^{-1}=\tau_1^q,\qquad \tau_1^e=1. \tag{6} \]
From (3)—(6) it follows that the elements \(\sigma_1,\tau_1,\gamma_{11}=\left(\dfrac{K_1/K_0}{\varepsilon_0}\right),\ldots,\gamma_{m1}=\left(\dfrac{K_1/K_0}{\varepsilon_m}\right)\) will form a system of generators of the group \(\Phi_1\). Therefore the mapping
\[ s\to\sigma_1,\qquad t\to\tau_1,\qquad a_1\to\gamma_{11},\ldots,\ a_m\to\gamma_{m1} \]
defines a homomorphism \(\mathfrak S\) of the group \(F\) onto \(\Phi_1\). Let \(F_0'\) be the Frattini subgroup of the group \(F_0\), and let \(F_1\) be the normal divisor of the group \(F\) generated by the groups \(F_0'\) and \(W\). Then \(F_1\) is the kernel of the homomorphism \(\mathfrak S\).
Next let \(F_0^{(n+1)}\) be the Frattini subgroup of the group \(F_0^{(n)}\), \(n=1,2,\ldots\), and let \(F_n\) be the normal divisor of the group \(F\) generated by the groups \(F_0^{(n)}\) and \(W\).
We choose inductively in \(\Phi_n\) representatives \(\sigma_n,\tau_n,\gamma_{1n},\ldots,\gamma_{mn}\) of the automorphisms \(\sigma_{n-1},\tau_{n-1},\gamma_{1,n-1},\ldots,\gamma_{m,n-1}\) so that the relations
\[ \tau_n^e=1,\qquad \sigma_n\tau_n\sigma_n^{-1}=\tau_n^q \]
are satisfied. The mapping
\[ s\to\sigma_n,\qquad t\to\tau_n,\qquad a_1\to\gamma_{1n},\ldots,\ a_m\to\gamma_{mn} \]
defines a homomorphism of the group \(F\) onto \(\Phi_n\), whose kernel coincides with \(F_n\). The group \(\Phi\) of the field \(K\) over \(k\) is the projective limit of the sequence
\[ \Phi_0\leftarrow \Phi_1\leftarrow \cdots \leftarrow \Phi_n\leftarrow\cdots \]
Therefore \(\Phi\) is isomorphic to the projective limit of the sequence
\[ F/F_0\leftarrow F/F_1\leftarrow \cdots \leftarrow F/F_n\leftarrow\cdots \]
The intersection of all \(F_n\) is equal to \(W\). The sequence \(\{F_n/W,\ n=0,1,2,\ldots\}\) forms a system of neighborhoods of the identity of the group \(F/W\), which consists of
all normal divisors \(N\) of the group \(F/W\) satisfying the conditions \(N \subseteq F_0/W\), and \([F_0/W;N]\) is equal to a power of \(p\). Theorem 1 follows from this.
An analogous result can be obtained in the case where \(k\) is any finite extension of the field \(R_p\), and \(K\) is the algebraic closure of the field.
Let \(K_0\) be the maximal extension of the field \(k\) without higher ramification. Over \(k\) the field \(K_0\) is generated by means of the totality of pairs of elements
\[ (\zeta_e,\sqrt[e]{\pi}), \]
where \(e\) runs through all natural numbers relatively prime to \(p\), and \(\zeta_e\) is a primitive \(e\)-th root of unity.
The group \(\Phi_0\) of the field \(K_0\) over \(k\) is the total completion of the group \(\Gamma\) with two generators \(\sigma\) and \(\tau\), defined as follows:
\[ \sigma=(\zeta_e\to\zeta_e^q,\ \sqrt[e]{\pi}\to\zeta_e\sqrt[e]{\pi}), \]
\[ \tau=(\zeta_e\to\zeta_e,\ \sqrt[e]{\pi}\to\zeta_e\sqrt[e]{\pi}); \]
\(\sigma\) and \(\tau\) satisfy the single relation \(\sigma\tau\sigma^{-1}=\tau^q\).
Iwasawa \((^4)\) showed that the group \(\Psi\) of the field \(K\) over \(K_0\) is a free topological \(p\)-group with a countable number of generators, and that the group \(\Phi\) of the field \(K\) over \(k\) is a semidirect extension of the group \(\Phi_0\) by means of \(\Psi\).
Moreover, Iwasawa showed that there exists a sequence of fields \(k_n,\ n=1,2,\ldots,\infty\), with the following properties:
-
\(k_n\) is a finite normal extension of the field \(k\) with simple ramification.
-
\[ \bigcup_{n=1}^{\infty} k_n=K_0. \]
-
Every principal unit \(\varepsilon_n\) of the field \(k_n\) has a representation
\[ \varepsilon_n=\varepsilon_{0n}^{a}\prod_{\substack{\nu=1\\ \eta\in\varphi_n}}^{m}\varepsilon_{\nu n}^{a_{\nu n}\eta^{i}}, \]
where \(\varepsilon_{0n},\varepsilon_{1n},\ldots,\varepsilon_{mn}\) is a system of principal units of the field \(k_n\); \(a\) and \(a_{\nu n}\) are integral \(p\)-adic numbers; \(\varphi_n\) is the group of the field \(k_n\) over \(k\).
- Let \(p^K\) be the maximal \(p\)-power order of a root of unity \(\zeta_K\) contained in \(K_0\). Then the representation (7) is unique up to \(p^K\)-th powers, and the congruences
\[ \varepsilon_{0n}^{\sigma}\equiv \varepsilon_{0n}^{g},\qquad \varepsilon_{0n}^{\tau}\equiv \varepsilon_{0n}^{h}, \pmod {p^K}\qquad\pmod {p^K} \]
hold, where \(g\) and \(h\) are determined by the equalities
\[ \zeta_K^{\sigma}=\zeta_K^{g},\qquad \zeta_K^{\tau}=\zeta_K^{h}. \]
- \(N_{k_l/k_n}(\varepsilon_{\nu n})=\varepsilon_{\nu l}\).
Let \(K_{1n}\) (respectively, \(K_1\)) be the union of all normal cyclic extensions of degree \(p^K\) of the field \(k_n\) (respectively, \(K_0\)). As above, we can compute the group \(\Phi_{1n}\) of the field \(K_{1n}\) over \(k\) and then pass to the projective limit of the sequence of groups \(\Phi_{11}\leftarrow\Phi_{12}\leftarrow\cdots\leftarrow\Phi_{1n}\leftarrow\cdots\).
Thus we obtain that the group of the field \(K_1\) over \(K_0\) has a minimal basis of the form
\[ \{\gamma_0',\ \eta'\gamma_\nu'\eta'^{-1};\ \nu=1,\ldots,m;\ \eta\in\Gamma\}, \]
where
\[ \gamma_\nu'=\lim_{n\to\infty}\left(\frac{K_{1n}/k_n}{\varepsilon_{\nu n}}\right) \]
and \(\eta'\) is the extension of the automorphism \(\eta\) to \(K_1\). \(\gamma_0'\) satisfies the relations
\[ \sigma'\gamma_0'\sigma'^{-1}=\gamma_0'^{\,g},\qquad \tau'\gamma_0'\tau'^{-1}=\gamma_0'^{\,h}. \]
Choose in \(\Phi\) representatives \(\bar{\sigma}, \bar{\tau}, \gamma_0, \ldots, \gamma_m\) of the automorphisms \(\sigma', \tau', \gamma'_0, \ldots, \gamma'_m\) so that the relations
\[ \bar{\sigma}\bar{\tau}\bar{\sigma}^{-1}=\bar{\tau}^{q}, \qquad \bar{\tau}\gamma_0\bar{\sigma}\bar{\tau}^{-1}=\gamma_0^h \]
are satisfied.
The relation \(\sigma'\gamma'_0\sigma'^{-1}=\gamma_0^{\prime g}\) extends to the relation
\[ \bar{\sigma}\gamma_0\bar{\sigma}^{-1}=\rho\gamma_0^g, \]
where \(\rho\in \Psi^{p^K}[\Psi,\Psi]\).
By the theorem on Burnside bases \((^5)\), the elements \(\gamma_0, \bar{\eta}\gamma_\nu\bar{\eta}^{-1}\), \(\nu=1,\ldots,m;\ \eta\in\Gamma\), form a minimal system of generators of the free topological \(p\)-group \(\Psi\). Consequently, the following holds.
Theorem 2. The group \(\Phi\) of the algebraic closure \(K\) of the field \(k\) is a semidirect extension of the group \(\Phi_0\) by means of the free topological \(p\)-group \(\Psi\) with minimal system of generators
\[ \{\gamma_0,\bar{\eta}\gamma_\nu\bar{\eta}^{-1};\ \nu=1,\ldots,m;\ \eta\in\Gamma\}. \]
The automorphism \(\bar{\sigma}\) (respectively \(\bar{\tau}\)) sends \(\gamma_0\) to \(\rho\gamma_0^g\) (respectively \(\gamma_0^h\)), where \(\rho\in \Psi^{p^K}[\Psi,\Psi]\).
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
23 XI 1960
REFERENCES
- I. R. Shafarevich, Matem. sborn., 20 (62), No. 2 (1947).
- Z. I. Borevich, Vestn. Leningradsk. univ., ser. matem., No. 19 (1956).
- M. Krasner, Acta Arithmetica, 3, No. 2 (1939).
- K. Iwasawa, Trans. Am. Math. Soc., 80, 448 (1955).
- H. Zassenhaus, Lehrbuch der Gruppentheorie, 1, 1937.