Polynomial with Galois Group $D_{2n}$ How does one construct a polynomial with Galois Group $D_{2n}$? A general method would be preferable or if that's impractical then an example of it being done for any n would be appreciated.
Thanks!
 A: One general method proceeds by making use of invariant polynomials.  Let $G$ be a candidate Galois group for an irreducible polynomial of degree $n$ over a field $F$ so that in particular $G$ has a transitive permutation action on $n$ objects $r_{i}$ which we identify with the roots of the polynomial.  Basic Galois theory then tells us that any polynomial in the $r_{i}$ which is invariant under the action of $G$ must then lie in $F$.  Thus $G$ stabilizes the invariant polynomial ring $F[r_{1}, \cdots, r_{n}]^{G}$.
Now, since $G$ is a subgroup of $S_{n}$ the invariant ring includes the elementary symmetric polynomials $\sigma_{i}$ defined as
$
\sigma_{1}=r_{1}+r_{2}+\cdots +r_{n},
$
$
\sigma_{2}=r_{1}r_{2}+r_{1}r_{3}+\cdots +r_{n-1}r_{n},
$
$\cdots = \cdots ,$
$\sigma_{n} = r_{1}r_{2}\cdots r_{n}. $
On the other hand in the splitting field, a polynomial with roots $r_{i}$ can be completely factored as
$\prod_{i}(z-r_{i}) = z^{n}-\sigma_{1}z^{n-1}+\sigma_{2}z^{n-2}+\cdots + (-1)^{n}\sigma_{n}.$
To construct a polynomial with Galois group $G$ we can then simply choose the $\sigma_{i}$ to be consistent with whatever relations occur in the invariant ring and write a polynomial as above.  In general this leads to a polynomial whose Galois group is a subgroup of $G$, but provided we choose the invariants sufficiently generically the Galois group will in fact be $G$ itself.  In general this method works well for groups of small order where the invariant rings are managable.
Now let us apply this to produce an example of a degree four polynomial over $\mathbb{Q}$ with Galois group $D_{4}$.  Up to a shift in the indeterminate we may assume that this polynomial takes the form
$
p(z)=z^{4}+\sigma_{2}z^{2}-\sigma_{3}z+\sigma_{4}.
$
In other words without loss of generality we may assume that the sum of the roots of $p(z)$ vanishes.
An elementary problem in the theory of finite group representations shows that the invariant ring of $D_{4}$ acting as permutation on four objects $r_{i}$ subject to the constraint that $\sum_{i}r_{i}=0$ is generated by four objects $\alpha, \beta, \chi, \lambda$ subject to the single relation $\alpha \lambda =\chi^{2}$.  Thus the relevant invariant polynomial ring is simply
$
\mathbb{Q}[r_{1}, r_{2}, r_{3}, -r_{1}-r_{2}-r_{3}]^{D_{4}}\cong \mathbb{Q}[\alpha, \beta, \chi, \lambda]/\langle \alpha \lambda=\chi^{2}\rangle.
$
Then the symmetric polynomials are expressed in terms of the generators of the invariant ring as
$
\sigma_{2}=-\frac{1}{8}(\beta+\alpha),
$
$\sigma_{3}= \frac{\chi}{16},$
$\sigma_{4}= \frac{1}{256}((\alpha-\beta)^{2}-\lambda).$
Thus any polynomial with Galois group $D_{4}$ can be written by choosing arbitrary $\alpha, \beta, \chi, \lambda$ subject to the single constraint in the invariant ring and plugging into the above.  For example, one solution with integer coefficients is given by
$
p(z)=z^{4}+z^{2}+2z+1.
$
A: This is done in:
MR1697454 (2000e:12013) 
Ledet, Arne(3-QEN)
Dihedral extensions in characteristic 0. (English, French summary) 
C. R. Math. Acad. Sci. Soc. R. Can. 21 (1999), no. 2, 46–52. 
12F12 (11R20) 
I am having trouble finding the actual paper, but if you write to the author, I am sure he can help out.
