Galois number fields with small root discriminant
Let \(K\) be a Galois extension of \({\mathbb Q}\) with (finite) group \(G\). For \(x\in K\), one has \(\sum_{g\in G} g(x)\in {\mathbb Q}\), and so there exists a map \(\text{tr}: K\rightarrow {\mathbb Q}\) defined by the rule \(x\mapsto\sum_{g\in G} g(x)\). This is the trace map of \(K/{\mathbb Q}\). There is a non-degenerate symmetric bilinear form \(B: K\times K\rightarrow {\mathbb Q}\) defined as \(B(x,y)=\text{tr}(xy)\) for \(x,y\in K\). Let \(R\) be the integral closure of \({\mathbb Z}\) in \(K\). Then \(B: R\times R\rightarrow {\mathbb Z}\). Since \(R\) is a finitely generated torsion-free module over the PID \({\mathbb Z}\), \(R\) is \({\mathbb Z}\)-free of rank \(N=| G|\). Let \(\{x_1,x_2,\dots,x_N\}\) denote a \({\mathbb Z}\)-basis for \(R\). Let \(A\) be the \(N\times N\) matrix whose \(i,j\)th entry is \(B(x_i,x_j)=\text{tr}(x_ix_j)\), \(1\leq i,j\leq N\). Then the discriminant of \(K\), \(D(K)\), is the ideal of \({\mathbb Z}\) generated by the non-zero integer \(\text{det}(A)\). We identify \(D(K)\) with the integer \(\text{det}(A)\). Certainly, \(D({\mathbb Q})=1\), and by Minkowski's Theorem, \(| D(K)| > 1\) whenever \(K\not = {\mathbb Q}\). The root discriminant of \(K\) is defined as \(d(K)=| D(K)|^{1\over N}\). Let \(C\geq 1\) be a real number, and let \(G\) be a finite group. Let \({\mathcal K}(G,C)\) denote the collection of all Galois extensions with group isomorphic to \(G\) for which \(d(K)\leq C\). For example, let \(n\geq 1\) be an integer, and let \(\zeta_n\) denote a primitive \(n\)th root of unity. For a rational prime \(p\), \(d({\mathbb Q}(\zeta_p))=p^{p-2\over p-1}<p\), and so, \({\mathbb Q}(\zeta_p)\in {\mathcal K}(C_{p-1},p)\) where \(C_{p-1}\) denotes the cyclic group of order \(p-1\). A natural question arises: Can one compute all of the field extensions in \({\mathcal K}(C_{p-1},p)\)? This is a reasonable question since for a fixed group \(G\) and fixed constant \(C\geq 1\), it is known that \({\mathcal K}(G,C)\) is finite. In the case that \(C\) is Serre's constant \(\Omega\) (\(\approx 44.7632\)), the authors in the paper under review state the following conjecture: Conjecture 1.1: \({\mathcal K}(G,\Omega)\) is empty for all but a finite number of \(G\). The focus of the paper under review is to provide evidence in support of Conjecture 1.1. If Conjecture 1.1 is true, then \({\mathcal K}(\Omega)\) is finite, where \({\mathcal K}(\Omega)\) denotes the collection of all Galois extensions \(K/{\mathbb Q}\) with \(d(K)\leq \Omega\). Moreover, it is plausible that \({\mathcal K}(\Omega)\) is finite since \(\bigcup_G{\mathcal K}(G,C)\) is finite for \(C<\Omega\) if we assume the Generalized Riemann Hypothesis. In order to test Conjecture 1.1, the authors give a formula for computing root discriminants. One has \(\displaystyle{d(K)=\prod_p p^{\beta_p}}\), for rational numbers \(\beta_p\). Note that the \(\beta_p\) are zero almost everywhere. The computation of \(\beta_p\not = 0\) depends on whether \(p\) is tamely ramified or not. If \(p\) is tamely ramified, the method is to write \(K={\mathbb Q}(\alpha)\) for some \(\alpha\in {\mathbb C}\). Let \(f(x)\) be the irreducible polynomial of \(\alpha\), and let \(f(x)=r_1(x)r_2(x)\cdots r_g(x)\) denote the factorization of \(f(x)\) over \({\mathbb Q}_p^{un}\), the maximal unramified extension of \({\mathbb Q}_p\). If \(e_i=\text{deg}(r_i(x))\), \(i=1,\dots,g\), and \(t=\text{lcm}(e_1,e_2,\dots,e_g)\), then \(\beta_p={t-1\over t}\). In the case that \(p\) is wildly ramified, there is a corresponding formula (though more complicated) which computes \(\beta_p\) using the slope data at \(p\). With the formula for \(d(K)\) the authors, using the PARI/GP system and the Atlas of Finite Groups, compute \(| {\mathcal K}(G,\Omega)|\) for each \(G\) abelian. They show that there are \(7063\) abelian extensions in \({\mathcal K}(\Omega)\) (Proposition 3.1). This shows that \({\mathcal K}(G,\Omega)\) is empty for all but finitely many abelian groups, which verifies Conjecture 1.1 in the case that \(G\) is abelian. Their method uses the fact that each abelian extension is a subfield of \({\mathbb Q}^{ab}\), the maximal abelian extension of \({\mathbb Q}\) in \({\mathbb C}\), which in turn, is the union of the subfields \({\mathbb Q}(\zeta_n)\) by the Kronecker-Weber Theorem. In the case that \(G\) is non-abelian, the authors compute \(| {\mathcal K}(G,\Omega)|\) for various cases of \(G\), including the symmetric groups \(S_3,S_4,S_5,S_6\), the alternating groups \(A_4,A_5,A_6\) and the dihedral groups \(D_4,D_5\). For example, the authors show that \({\mathcal K}(A_6,\Omega)\) contains \(5\) fields, and that \({\mathcal K}(S_6,\Omega)\) contains \(13\) fields. For each of these fields, they give the value of the Galois root discriminant and the slope data, and find a defining polynomial for the field (Tables 5.2, 5.3).
scientific article; zbMATH DE number 5526764
scientific article