On the norm groups of Galois \(2\mathfrak n\)-extensions of algebraic number fields. (English) Zbl 1176.11056
Let \(K\) be a finite extension of \({\mathbb Q}\), let \(L\) be a finite extension of \(K\), and let \(\overline {\mathbb Q}\) denote an algebraic closure of \({\mathbb Q}\). Let \(\text{Em}(L/K)\) denote the set of embeddings \(\tau: L\rightarrow \overline {\mathbb Q}\) which fix \(K\). The norm \(N_{L/K}: L\rightarrow K\) is defined as \(x\mapsto \prod_{\tau\in \text{Em}(L/K)} \tau(x)\), \(x\in L\).
Let \(\{|\;|_\nu\}_{\nu\in W}\) be a set of representatives for the equivalence classes of normalized absolute values on \(L\). For each \(\nu\in W\), let \(L_\nu\) denote the completion of \(L\) with respect to the \(|\;|_\nu\)-topology on \(L\). For each discrete \(|\;|_\nu\), the ring of integers \(S_\nu\) of \(L_\nu\) is a compact subset of \(L_\nu\). Let \(\{Y_\nu\}_{\nu\in W}\) denote the family of subsets of \(L_\nu\) defined as \(Y_\nu=S_\nu\) if \(\nu\) is discrete and \(Y_\nu=L_\nu\), otherwise. The adele ring \(V_L\) of \(L\) is the ring \(\prod_{\nu\in W} L_\nu\) together with the restricted product topology on \(\prod_{\nu\in W} L_\nu\) with respect to the family \(\{Y_\nu\}\). There is an injection of commutative rings \(L\rightarrow V_L\).
Let \((1_\nu)\) be defined as \(1_\nu=1\) for all \(\nu\in W\). Then \((1_\nu)\) is in \(V_L\), and so, \(V_L\) is a commutative ring with unity. The multiplicative group of units of \(V_L\) is the idele group of \(L\), denoted by \(I_L\). The norm map \(N_{L/K}\) can be defined on the \(I_L\) since for each \(\nu\in W\) lying above a prime \(Q\in K\), the set \((\text{Em}(L/K))_\nu=\{\tau\in \text{Em}(L/K):\;\tau(\nu)=\nu\}\) is a subset of \(\text{Em}(L/K)\). Note that \(\tau\in (\text{Em}(L/K))_\nu\) is an embedding \(\tau: L_\nu\rightarrow \overline {{\mathbb Q}_q}\) which fixes \(K_Q\). Here \(q\) is the rational prime which lies below \(Q\). In this manner \(N_{L/K}: I_L\rightarrow V_K\).
The map \(C: I_L\rightarrow {\mathbb R}^+\) defined as \(C((a_\nu ))=\prod_{\nu\in W}| a_\nu|_\nu\) is a homomorphism of multiplicative groups, called the content map. One has that \(L^*\) injects into \(I_L^1\), the kernel of the content map. Thus \(N_{L/K}(L^*)=K^*\cap N_{L/K}(L^*)\subseteq K^*\cap N_{L/K}(I_L)\). Put \(N(L/K)=K^*\cap N_{L/K}(I_L)\). Then the quotient group \(N(L/K)/N_{L/K}(L^*)\) is the total obstruction to the Hasse Norm Principle (HNP) for \(L/K\), that is, if \(N(L/K)/N_{L/K}(L^*)\) is trivial, then \(L/K\) satisfies the HNP. This is the context of the paper under review.
Let \(p\) be a rational prime \(p\geq 3\), let \(C_p=\langle \sigma\rangle\) be the cyclic group of order \(p\). Let \(Z\) be the \(2n\)-group defined as \(Z=\langle a_1,a_2,\dots,a_{p-1},\sigma\rangle\) subject to the relations \(a_i^2=\sigma^p=1\), \(a_i^{-1}a_j^{-1}a_ia_j=1\), \(1\leq i,j\leq p-1\), \(\sigma a_i=a_{i+1}\sigma\), \(1\leq i\leq p-2\), and \(\sigma a_{p-1}=a_1a_2\cdots a_{p-1}\sigma\).
Let \(A=\langle a_1,a_2,\dots,a_{p-1}\rangle\), \(B_1=\langle a_2,a_3,\dots,a_{p-1}\rangle\). Each nontrivial element of \(A\) can be written in reduced form as the string \(a_{i_1}a_{i_2}\cdots a_{i_m}\)for integers \(1\leq i_1 < i_2 < \cdots < i_m\leq p-1\), \(1\leq m\leq p-1\). Let \(B_2\) be the subgroup of \(A\) generated by strings of this form with \(m\) even.
Let \(D/P\) be a Galois extension of number fields with group \(Z\), let \(K\) be the fixed field of \(A\), let \(F_1\) be the fixed field of \(B_1\) and let \(F_2\) be the fixed field of \(B_2\). One has \(\text{Gal}(D/K)=A\), \(\text{Gal}(D/F_1)=B_1\) and \(\text{Gal}(D/F_2)=B_2\). Also, \(K/P\) is a Galois extension with group \(Z/A\cong C_p\).
We have the norm \(N_{K/P}: K\rightarrow P\) defined as \(x\mapsto \prod_{\sigma\in C_p} \sigma(x)\), and other norm maps \(N_{F_1/P}: F_1\rightarrow P\) and \(N_{F_2/P}: F_2\rightarrow P\) defined over collection of embeddings \(Em(F_1/P)\), \(Em(F_2/P)\) as above. Since \(\text{Gal}(K/P)=C_p\), \(K/P\) satisfies the HNP, that is, \(N(K/P)=N_{K/P}(K^*)\) where \(N(K/P)=P^*\cap N_{K/P}(I_K)\).
One of the key results of the paper under review (Theorem 1.6) is that for \(p\geq 5\), \[ N_{F_1/P}(F_1^*)=N_{F_2/P}(F_2^*)=N_{K/P}(K^*). \] This is achieved by showing that \(N(K/P)=N(F_1/P)=N(F_2/P)\) and that \(F_1/P\) and \(F_2/P\) satisfy the HNP. Theorem 1.6 is used to prove the main theorem of the paper (Theorem 2.8) which states that if \(K/k\) is a Galois extension whose group is a \(2n\)-group, then the interval \((N_{K/k}(K^*),k^*)\) contains an infinite number of norm groups.
We sketch the method of proof. Let \(K/k\) be a Galois extension of number fields whose group \(\tilde G\) is a \(2n\)-group. Necessarily, \(\tilde G\) contains a \(2m\)-subgroup \(S\) which is one of two specific types, both containing the cyclic group of order \(p\), \(C_p\) (see Proposition 2.1). The pair \((S,\tilde G)\) determines a group \(G\), a short exact sequence \(1\rightarrow N\rightarrow G\rightarrow \tilde G\rightarrow 1\) and a finite extension \(E/k\) for which \(\text{Gal}(E/k)=G\), \(K\subseteq E\) and \(NC_p\leq G\). This construction also yields a finite extension \(L/k\), \(L\subseteq E\). Let \(P\) be the fixed field of \(NC_p\), so that \(\text{Gal}(E/P)=NC_p\). There is a short exact sequence \(1\rightarrow C\rightarrow NC_p\rightarrow Z\rightarrow 1\). Let \(D\) be the fixed field of \(C\), so that \(\text{Gal}(D/P)=Z\). Now by Theorem 1.6, \(N_{F_1/P}(F_1^*)=N_{F_2/P}(F_2^*)=N_{K/P}(K^*)\), which by Theorem 2.5 yields the strict containment \(N_{K/k}(K^*)\subset N_{L/k}(L^*) \subset k^*\).
Moreover, suppose \(E'/k\) and \(E''/k\) are field extensions with group \(G\) and \(E'\cap E''=K\). Then the subfields \(L'/k\) and \(L''/k\) obtained as above have the property that \(N_{L'/k}({L'}^*)\not = N_{L''/k}({L''}^*)\) (Proposition 2.6). The author then shows that there are an infinite number of such \(L/k\) (Proposition 2.7). This gives the main result.
Let \(\{|\;|_\nu\}_{\nu\in W}\) be a set of representatives for the equivalence classes of normalized absolute values on \(L\). For each \(\nu\in W\), let \(L_\nu\) denote the completion of \(L\) with respect to the \(|\;|_\nu\)-topology on \(L\). For each discrete \(|\;|_\nu\), the ring of integers \(S_\nu\) of \(L_\nu\) is a compact subset of \(L_\nu\). Let \(\{Y_\nu\}_{\nu\in W}\) denote the family of subsets of \(L_\nu\) defined as \(Y_\nu=S_\nu\) if \(\nu\) is discrete and \(Y_\nu=L_\nu\), otherwise. The adele ring \(V_L\) of \(L\) is the ring \(\prod_{\nu\in W} L_\nu\) together with the restricted product topology on \(\prod_{\nu\in W} L_\nu\) with respect to the family \(\{Y_\nu\}\). There is an injection of commutative rings \(L\rightarrow V_L\).
Let \((1_\nu)\) be defined as \(1_\nu=1\) for all \(\nu\in W\). Then \((1_\nu)\) is in \(V_L\), and so, \(V_L\) is a commutative ring with unity. The multiplicative group of units of \(V_L\) is the idele group of \(L\), denoted by \(I_L\). The norm map \(N_{L/K}\) can be defined on the \(I_L\) since for each \(\nu\in W\) lying above a prime \(Q\in K\), the set \((\text{Em}(L/K))_\nu=\{\tau\in \text{Em}(L/K):\;\tau(\nu)=\nu\}\) is a subset of \(\text{Em}(L/K)\). Note that \(\tau\in (\text{Em}(L/K))_\nu\) is an embedding \(\tau: L_\nu\rightarrow \overline {{\mathbb Q}_q}\) which fixes \(K_Q\). Here \(q\) is the rational prime which lies below \(Q\). In this manner \(N_{L/K}: I_L\rightarrow V_K\).
The map \(C: I_L\rightarrow {\mathbb R}^+\) defined as \(C((a_\nu ))=\prod_{\nu\in W}| a_\nu|_\nu\) is a homomorphism of multiplicative groups, called the content map. One has that \(L^*\) injects into \(I_L^1\), the kernel of the content map. Thus \(N_{L/K}(L^*)=K^*\cap N_{L/K}(L^*)\subseteq K^*\cap N_{L/K}(I_L)\). Put \(N(L/K)=K^*\cap N_{L/K}(I_L)\). Then the quotient group \(N(L/K)/N_{L/K}(L^*)\) is the total obstruction to the Hasse Norm Principle (HNP) for \(L/K\), that is, if \(N(L/K)/N_{L/K}(L^*)\) is trivial, then \(L/K\) satisfies the HNP. This is the context of the paper under review.
Let \(p\) be a rational prime \(p\geq 3\), let \(C_p=\langle \sigma\rangle\) be the cyclic group of order \(p\). Let \(Z\) be the \(2n\)-group defined as \(Z=\langle a_1,a_2,\dots,a_{p-1},\sigma\rangle\) subject to the relations \(a_i^2=\sigma^p=1\), \(a_i^{-1}a_j^{-1}a_ia_j=1\), \(1\leq i,j\leq p-1\), \(\sigma a_i=a_{i+1}\sigma\), \(1\leq i\leq p-2\), and \(\sigma a_{p-1}=a_1a_2\cdots a_{p-1}\sigma\).
Let \(A=\langle a_1,a_2,\dots,a_{p-1}\rangle\), \(B_1=\langle a_2,a_3,\dots,a_{p-1}\rangle\). Each nontrivial element of \(A\) can be written in reduced form as the string \(a_{i_1}a_{i_2}\cdots a_{i_m}\)for integers \(1\leq i_1 < i_2 < \cdots < i_m\leq p-1\), \(1\leq m\leq p-1\). Let \(B_2\) be the subgroup of \(A\) generated by strings of this form with \(m\) even.
Let \(D/P\) be a Galois extension of number fields with group \(Z\), let \(K\) be the fixed field of \(A\), let \(F_1\) be the fixed field of \(B_1\) and let \(F_2\) be the fixed field of \(B_2\). One has \(\text{Gal}(D/K)=A\), \(\text{Gal}(D/F_1)=B_1\) and \(\text{Gal}(D/F_2)=B_2\). Also, \(K/P\) is a Galois extension with group \(Z/A\cong C_p\).
We have the norm \(N_{K/P}: K\rightarrow P\) defined as \(x\mapsto \prod_{\sigma\in C_p} \sigma(x)\), and other norm maps \(N_{F_1/P}: F_1\rightarrow P\) and \(N_{F_2/P}: F_2\rightarrow P\) defined over collection of embeddings \(Em(F_1/P)\), \(Em(F_2/P)\) as above. Since \(\text{Gal}(K/P)=C_p\), \(K/P\) satisfies the HNP, that is, \(N(K/P)=N_{K/P}(K^*)\) where \(N(K/P)=P^*\cap N_{K/P}(I_K)\).
One of the key results of the paper under review (Theorem 1.6) is that for \(p\geq 5\), \[ N_{F_1/P}(F_1^*)=N_{F_2/P}(F_2^*)=N_{K/P}(K^*). \] This is achieved by showing that \(N(K/P)=N(F_1/P)=N(F_2/P)\) and that \(F_1/P\) and \(F_2/P\) satisfy the HNP. Theorem 1.6 is used to prove the main theorem of the paper (Theorem 2.8) which states that if \(K/k\) is a Galois extension whose group is a \(2n\)-group, then the interval \((N_{K/k}(K^*),k^*)\) contains an infinite number of norm groups.
We sketch the method of proof. Let \(K/k\) be a Galois extension of number fields whose group \(\tilde G\) is a \(2n\)-group. Necessarily, \(\tilde G\) contains a \(2m\)-subgroup \(S\) which is one of two specific types, both containing the cyclic group of order \(p\), \(C_p\) (see Proposition 2.1). The pair \((S,\tilde G)\) determines a group \(G\), a short exact sequence \(1\rightarrow N\rightarrow G\rightarrow \tilde G\rightarrow 1\) and a finite extension \(E/k\) for which \(\text{Gal}(E/k)=G\), \(K\subseteq E\) and \(NC_p\leq G\). This construction also yields a finite extension \(L/k\), \(L\subseteq E\). Let \(P\) be the fixed field of \(NC_p\), so that \(\text{Gal}(E/P)=NC_p\). There is a short exact sequence \(1\rightarrow C\rightarrow NC_p\rightarrow Z\rightarrow 1\). Let \(D\) be the fixed field of \(C\), so that \(\text{Gal}(D/P)=Z\). Now by Theorem 1.6, \(N_{F_1/P}(F_1^*)=N_{F_2/P}(F_2^*)=N_{K/P}(K^*)\), which by Theorem 2.5 yields the strict containment \(N_{K/k}(K^*)\subset N_{L/k}(L^*) \subset k^*\).
Moreover, suppose \(E'/k\) and \(E''/k\) are field extensions with group \(G\) and \(E'\cap E''=K\). Then the subfields \(L'/k\) and \(L''/k\) obtained as above have the property that \(N_{L'/k}({L'}^*)\not = N_{L''/k}({L''}^*)\) (Proposition 2.6). The author then shows that there are an infinite number of such \(L/k\) (Proposition 2.7). This gives the main result.
Reviewer: Robert Underwood (Tallahassee)
References:
| [1] | Alperin, J.; Lyons, R., On conjugacy classes of \(p\)-elements, J. Algebra, 19, 536-537 (1971) · Zbl 0238.20026 |
| [2] | Drakokhrust, Yu. A.; Platonov, V. P., The Hasse norm principle for algebraic number fields, Izv. Akad. Nauk SSSR. Izv. Akad. Nauk SSSR, Math. USSR Izv., 29, 299-322 (1987), English translation: · Zbl 0624.12009 |
| [3] | Fein, B.; Kantor, W. M.; Schacher, M., Relative Brauer groups II, J. Reine Angew. Math., 328, 39-57 (1981) · Zbl 0457.13004 |
| [4] | Feit, W.; Thompson, J. G., Solvability of groups of odd order, Pacific J. Math., 13, 3, 775-1029 (1963) · Zbl 0124.26402 |
| [5] | Guralnick, R.; Stern, L., Solitary Galois extensions of algebraic number fields, J. Number Theory, 50, 1-32 (1995) · Zbl 0822.11075 |
| [6] | Neukirch, J., Über das Einbettungsproblem der Algebraischen Zahlentheorie, Invent. Math., 21, 59-116 (1973) · Zbl 0267.12005 |
| [7] | Neukirch, J., Algebraic Number Theory, Grundlehren Math. Wiss., vol. 322 (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0956.11021 |
| [8] | Stern, L., On the norm groups of global fields, J. Number Theory, 32, 203-219 (1989) · Zbl 0687.12006 |
| [9] | Stern, L., On the equality of norm groups of global fields, J. Number Theory, 36, 108-126 (1990) · Zbl 0718.11056 |
| [10] | Stern, L., Criterion for the equality of norm groups of idele groups of algebraic number fields, J. Number Theory, 62, 338-352 (1997) · Zbl 0869.11091 |
| [11] | Stern, L., On an obstruction to the Hasse norm principle and the equality of norm groups of algebraic number fields, J. Number Theory, 75, 237-261 (1999) · Zbl 0938.11054 |
| [12] | Suzuki, M., Group Theory I, Grundlehren Math. Wiss., vol. 247 (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0472.20001 |
| [13] | Weiss, E., Cohomology of Groups (1969), Academic Press: Academic Press New York · Zbl 0192.34204 |
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