The Split, Abelian Kernel, Case of the Regular Inverse Galois Problem

This file is an addendum to RIGP.html. When an extension splits, and has abelian kernel, some elements of the regular inverse Galois problem seem easy. Using [FrJ, §16.4]₂, we comment on when this is encouraging – as a guide for the Nielsen-RIGP conjecture – and, equally, when it is misleading – because it doesn't hint at problems in extending explicit realizations to where the kernel is nilpotent, or with the non-split case.

I. NOTATION FOR SEMI-DIRECT AND WREATH PRODUCTS:

Consider this short exact sequence of groups, which we assume is split:
Basic Sequence: 1 →Ker →G→G^*→1.

That is, you can represent G as 2x2 matrices of the form:

|g* 0 |
| a 1 |, with g*∈G^* and a ∈Ker

and where matrix multiplication gives the multiplication in G. If we call the matrix above M( g*, a) and we multiply it by M( h*, b), then the result is what you would expect from matrix multiplication:

M(g*h*, (a)h*+b), as if h* is a matrix acting on the right of a.

We write G as Ker×^s G^* – the semi-direct product of Ker and G^*. If instead you wanted a left action of G^* on Ker, you could put a in the upper right corner, and 0 in the lower left. (I've written the second slot with a "+" as if Ker is abelian. That, however, is unnecessary; we could write that instead as (a)h*^. b.) The former case is that of [FrJ, p. 251]₂, though it is common to see the other. The notation with G^* on the right makes transparent our homomorphism G→G^*.

Now suppose T: G^*→S_n is a degree n permutation representation of G^*. Let M be any group whatsoever. Then, a natural case of semi-direct product takes Ker = Mⁿ, the direct product of M, n times. The action of an element g*∈G^* is just to permute the slots of Ker according to the permutation that g* represents. The resulting semi-direct product is called the wreath product. Compatible with [FrJ, p. 252]₂ we could denote it M wr_T G^*.

II. REALIZING WREATH PRODUCTS REGULARLY:

Assume we have regular realizations of G^*and M. Then, [FrJ, Lem. 13.8.1]₂ gives a regular realization of M wr_T G^*= G. Indeed, this old construction uses the following idea. Corresponding to T, there is a degree n extension L/Q(z) of Q(z) inside the regular Galois extension L^{^}/Q(z) having G^* as group. If c is a primitive element (generator) for it over Q(z), then let c₁,…, c_n, be the 0th, …, n-1st powers of it, giving a basis of L over Q(z).

Now consider the linear form v=∑_i=1ⁿ c_iw_i, with the w_i_'s algebraically independent of any other variable. Denote G( L^{^}/L) by G^*(1). Elements of each (right) coset G^*(1)h, h ∈ H, produce a new linear form by acting on the right of the c_i_'s (not touching the w_i_'s). Label the resulting n distinct linear forms v₁,…, v_n. The matrix of the equation – for solving for the w_i_'s in terms of the v_i_'s – is a famous invertible matrix – the discriminant – from algebraic number theory.

Now let f(z,y) give a regular realization of M over Q(z). Plug in any v_i for z. Then, the splitting fields for all the f(v_i,y)'s, over Q(w₁,…, w_n), composite to a regular extension with group G. The argument is almost identical to that produced in [FrJ, § 10.6]₂, famously referred to as Weil's restriction of scalars.

The gist here – as Ax showed me in 1968 – starts with this assumption. Suppose you have an absolutely irreducible curve C over a degree n extension K/Q. Then the restriction of scalars applies to the curve to give an absolutely irreducible variety W over Q which has a Q point if and only if C has a K point. The conclusion: Any algebraic extension of a P(seudo)A(lgebraically)C(losed) field is also PAC. That is, whether K is PAC reduces to a simple test: Do absolutely irreducible curves over the base field have K rational points. The [FrJ]₂ quotation is to an unpublished manuscript of Roquette from 1975.

III. REALIZING Ker×^s G^* REGULARLY WHEN Ker IS ABELIAN:

Again, consider the Basic Sequence of §I, where Ker is abelian. Reduce easily, to where Ker is an indecomposable (not a direct sum of smaller G^* modules) p group whose exponent we take to be q. That is, q is the minimal integer for which q^.Ker = {0}. The first step is to realize Ker as a regular extension of Q(z) without considering the G^* action. With no loss we can take Ker, which is a direct sum of cyclic groups, to be Z/n.

Quotient Principle: A regular realization of G automatically gives a regular realization for its quotient G^*.

Abelian Split Corollary: Suppose U is a Z/n[G^*] module. Then, U is a quotient of a sum of copies of the regular representation of G^*. In particular, U×^s G^* has natural regular realizations.

Proof: As an abelian group Z/n[G^*] is a direct sum of several copies of the module M=Z/n. Therefore, given a regular realization of M, §II gives a regular realization of Z/n[G^*]×^s G^*=M wr_T G^* with T the regular representation.

Take K=Q and apply the BCL in this simplest case. It says, if you use only n-cycles, the minimal number of branch points you can possibly use to regularly realize M is φ(n)=|G(Q(e^2π
i/n/Q)|. Further, to realize that case you must take the branch points as the conjugates of a primitive generator of Q(e^2π
i/n/Q). [FrJ, Lem.~16.3.1]₂ does that explicitly in what amounts to a very old construction for the regular realization of M.

That gives us branch cycles for M, and so for the regular reallization of Z/n[G^*]^t×^s G^* for any value of t. Now use that for some t there is a group cover Z/n[G^*]^t×^s G^*→ U×^s G^* to conclude from The Quotient Principle how to construct regular realizations of U×^s G^*.

Example [Dihedral Groups]: The dihedral group case D_p^k+1 is the Basic sequence when G^*= Z/2= ± 1 and Ker = Z/p^k+1, and p is odd. Then, the regular representation, Z/p^k+1[G^*] is a direct sum of the trivial representation (-1 acts trivially on Ker) and the representation on Ker where -1 acts by a∈Ker → -a. Similarly, whenever the module Ker is indecomposable, and |Ker| is prime to |G^*|, we can be very explicit on the way the proof of the corollary produces C'_Ker.

[FrJ] M. Fried and M. Jarden, Field Arithmetic, Springer Ergebnisse II Vol 11 (1986) 455 pgs., 2nd edition 2004, 780 pps. ISBN 3-540-22811-x. We quote here both the first and second ed., using [FrJ]₁ and [FrJ]₂ respectively.