Ten years later, Erbach, Fischer and McKay [EFM] published the trinomial

In fact, I showed:

**
Theorem (NDE 1999): Trinomials
ax^{7} + bx + c
over a field K of characteristic zero
whose Galois group is contained in G_{168}
are parametrized by the curve
**

A search for rational points on this curve finds seven,
with *x*=0, -3, 1/9, and Infinity.
Of these, the two points above *x*=-3,
and one of the points with *x*=1/9,
yield degenerate trinomials.
The Trinks-Matzat trinomial comes from the Weierstrass point x=0;
one of the two points at infinity
yields the Erbach-Fischer-McKay trinomial.
The two new examples come from the other point at infinity
and the nondegenerate point with *x*=1/9.

Now *C*_{168} is a curve of genus 2.
As is well known, Mordell conjectured, and Faltings proved [F1,F2],
that every algebraic curve of genus at least 2
has only finitely many points over any number field.
In our case, that means that there are
only finitely many equivalence classes of
*G*_{168}-trinomials over any number field.

But it can be much harder to list all cases
than to prove that the list is finite.
None of the known proofs of the Mordell conjecture
provides an effective procedure for finding all the rational points
and proving that the list is complete, even in the first case
of a genus-2 curve over **Q**. In recent years,
much effort has gone into solving this problem in practice,
and ever more curves are yielding to the combination of
theoretical insight and computational power; see for instance [B].

Applying these tools to *C*_{168},
Nils Bruin proved:

**
Theorem (N.Bruin, 2001):
There are no more rational points on this curve.
**

Therefore, every trinomial
*ax*^{7} + *bx* + *c*
over **Q** whose Galois group is contained in
*G*_{168} is equivalent to one of the four trinomials
displayed above.

Similar analysis applies to trinomials
*ax ^{n}*+

(attributed by Matzat [M2, pages 90-91 (Satz 3)] to Weber
[W, section 189]). Moreover, the Galois group is contained
in the 10-element dihedral group if and only if
*u*=*t*-1/*t*, when the quintic
is equivalent to

this is attributed by Matzat [M2, p.93 (Satz 4)] to [JRYZ].
Likewise, if *n*=6, we can ask that *G* be
the transitive 120- or 60-element subgroup of *S*_{6}.
(These are the images of the obvious subgroups *S*_{5}
and *A*_{5} under an outer automorphism of
*S*_{6}; they can also be obtained as
PGL_{2}(**Z**/5**Z**) and
PSL_{2}(**Z**/5**Z**)
acting on the six points of the projective line over
**Z**/5**Z**.)
Here the general such sextic is

for the 120-element group; for the 60-element group,
take *u*=*t*^{2}. Matzat reports
this on page 94 of [M2], and attributes it, together with
the formulas for sextic trinomials for several other
(imprimitive but) transitive subgroups of *S*_{6},
to Malle [M1]. (Matzat also mentions the genus-2 curve we call
*C*_{168} on page 95, but does not exhibit equations
or points on this curve, and gives its genus incorrectly as 3.)

The first possibility with *k*>1 is
*n*=5 and *k*=2. We found
** (but see
addendum below)** :

**
Theorem (NDE 1999): Trinomials
ax^{5} + bx^{2} + c
over a field K of characteristic zero
whose Galois group is contained in the 20-element subgroup
of S_{5} are parametrized by the elliptic curve
**

This is an elliptic curve of conductor 15,
labelled 15F in Tingley's ``Antwerp'' tables and 15-A4 by Cremona.
Over **Q**, this curve has rank zero,
so that there are only finitely many equivalence classes
of trinomials *ax*^{5}+*bx*^{2}+*c*
with solvable Galois groups. More specifically,
*C*_{20}(**Q**) is cyclic of order 8;
three of its points (the origin and the generators (2,6) and (32,171))
yield degenerate quintics, three produce trinomials with dihedral
Galois group, and the remaining two
(corresponding to the other two points not divisible by 2
in *C*_{20}(**Q**))
produce quintics with 20-element Galois group:

`
Added 8/2002:
Blair K. Spearman and Kenneth S. Williams draw my attention
to their paper [SW], which obtains all the solvable quintic trinomials
using the sextic resolvent of a quintic. For trinomials
X^{5}+aX+b,
their work is anticipated by [W] and [JRYZ].
For X^{5}+aX^{2}+b,
they in turn anticipated my 1999 computation.
There are a few minor differences: besides the alternative approach
to determining the curve C_{20}, Spearman and Williams
compute C_{20}(Q) themselves
by carrying out a 2-descent rather than citing previous work
on the arithmetic of that curve; and they give explicit solutions
by radicals of each of the five quintics, but do not apparently notice
that two of them generate the same field.
`

For our final example, take *n*=8 and *k*=1,
and let *G* be the group *G*_{1344}
of affine linear transformations of
(**Z**/2**Z**)^{3}.
[This is the semidirect product of *G*_{168} with
(**Z**/2**Z**)^{3};
equivalently, *G*_{1344} is the group of invertible
4-by-4 matrices over **Z**/2**Z**
whose bottom row is 0001, acting on the 8 column vectors
of height 4 with last coordinate 1.]
We then have:

**
Theorem (NDE 1999): Trinomials
ax^{8} + bx + c
over a field K of characteristic zero
whose Galois group is contained in G_{1344}
are parametrized by the curve
**

Current ``technology'' is not yet up to provably listing
all the **Q**-points of *C*_{1344}.
Still, we can search for points of small height, i.e., whose
*x*-coordinates have small numerator and denominator;
the larger the height, the harder it is for
2*x*^{6}+4*x*^{5}+36*x*^{4}+16*x*^{3}-45*x*^{2}+190*x*+1241
to be a perfect square. We found a total of eight points
with numerator and denominator of at most three digits:
the four pairs with *x*=2, 1, -1, and -5.
Eliminating three degenerate polynomials,
we recovered the polynomials

All of these degree-8 trinomials seem to be new.
Michael Stoll extended the search for rational points
on *C*_{1344} from three to five digits,
and found none besides the four pairs already known.
We naturally conjecture that there are no further rational points
on *C*_{1344}, and thus that every trinomial
*ax*^{8}+*bx*+*c*
with Galois group contained in *C*_{1344}
is equivalent to one of the four displayed above.

[EFM] D.W. Erbach, J. Fischer, and J. McKay: Polynomials with Galois group PSL(2,7),

[M1] G. Malle: Polynomials for primitive nonsolvable permutation groups of degree

[M2] B.H. Matzat:

[F1] G. Faltings: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern,

[F2] G. Faltings: Diophantine approximation on Abelian varieties,

[JRYZ] G. Roland, N. Yui, D. Zagier: A parametric family of quintic polynomials with Galois group D

[SW] B.K. Spearman, K.S. Williams: On solvable quintics

[W] H. Weber: