Masanari Kida (UEC, Tokyo Japan)
Classification of Brumer's quintic dihedral polynomials

Abstract: It is known that the Galois group of the quintic polynomial
X^5 + (-3+a) X^4 + (3+b-a)X^3 + (-1-a -2b+a^2)X^2  +bX +a
over Q(a,b) is isomorphic to the dihedral group D_5 of order 10.
Moreover, it is generic for D_5. In particular, every D_5 extension
over Q can be obtained by specializing the parameters a and b.
In this talk, I will discuss when it is irreducible and when two of
these define isomorphic fields. The classification closely relates
to the Mordell-Weil group of certain elliptic curve.
This method also can be applied for generic D_3 polynomials.
This work is a joint work with Atsushi Sato and Yuichi Rikuna.